%%% -*-BibTeX-*- %%% ==================================================================== %%% BibTeX-file{ %%% author = "Nelson H. F. Beebe", %%% version = "1.16", %%% date = "12 May 2015", %%% time = "06:07:18 MDT", %%% filename = "toct.bib", %%% address = "University of Utah %%% Department of Mathematics, 110 LCB %%% 155 S 1400 E RM 233 %%% Salt Lake City, UT 84112-0090 %%% USA", %%% telephone = "+1 801 581 5254", %%% FAX = "+1 801 581 4148", %%% URL = "http://www.math.utah.edu/~beebe", %%% checksum = "55811 3145 17894 162098", %%% email = "beebe at math.utah.edu, beebe at acm.org, %%% beebe at computer.org (Internet)", %%% codetable = "ISO/ASCII", %%% keywords = "bibliography, BibTeX, ACM Transactions on %%% Computation Theory", %%% license = "public domain", %%% supported = "no", %%% docstring = "This is a COMPLETE BibTeX bibliography for %%% the journal ACM Transactions on Computation %%% Theory (CODEN ????, ISSN 1942-3454), %%% covering all journal issues from 2009 -- %%% date. %%% %%% The journal has a World-Wide Web site at: %%% %%% http://www.acm.org/pubs/toct %%% %%% Tables-of-contents are available at: %%% %%% http://www.acm.org/pubs/contents/journals/toct/ %%% http://portal.acm.org/browse_dl.cfm?idx=J1190 %%% %%% Qualified subscribers can retrieve the full %%% text of recent articles in PDF form. %%% %%% At version 1.16, the COMPLETE journal %%% coverage looked like this: %%% %%% 2009 ( 7) 2012 ( 15) 2015 ( 5) %%% 2010 ( 5) 2013 ( 18) %%% 2011 ( 6) 2014 ( 16) %%% %%% Article: 72 %%% %%% Total entries: 72 %%% %%% The initial draft was extracted from the %%% ACM Web site, with manual corrections and %%% additions from bibliographies in the TeX %%% User Group collection, the author's %%% personal bibliography files, the Compendex %%% database, and a very large computer science %%% bibliography collection on ftp.ira.uka.de %%% in /pub/bibliography to which many people %%% of have contributed. Where multiple %%% sources of a particular entry existed, %%% field values have been manually merged to %%% preserve maximal information. Missing %%% entries were identified by software %%% developed for the TeX User Group and BibNet %%% bibliography archive projects, and were %%% then supplied from the original journal %%% issues. Questions arising from conflicting %%% data were resolved by consulting the %%% original journal issues. %%% %%% ACM copyrights explicitly permit abstracting %%% with credit, so article abstracts, keywords, %%% and subject classifications have been %%% included in this bibliography wherever %%% available. Article reviews have been %%% omitted, until their copyright status has %%% been clarified. %%% %%% The bibsource keys in the bibliography %%% entries below indicate the data sources, %%% usually the Karlsruhe computer science %%% bibliography archive for the first two %%% volumes, or the journal Web site or the %%% Compendex database, both of which lack %%% coverage of this journal before 1985. %%% %%% URL keys in the bibliography point to %%% World Wide Web locations of additional %%% information about the entry. %%% %%% Spelling has been verified with the UNIX %%% spell and GNU ispell programs using the %%% exception dictionary stored in the %%% companion file with extension .sok. %%% %%% BibTeX citation tags are uniformly chosen %%% as name:year:abbrev, where name is the %%% family name of the first author or editor, %%% year is a 4-digit number, and abbrev is a %%% 3-letter condensation of important title %%% words. Citation tags were automatically %%% generated by software developed for the %%% BibNet Project. %%% %%% In this bibliography, entries are sorted in %%% publication order, using ``bibsort -byvolume.'' %%% %%% The checksum field above contains a CRC-16 %%% checksum as the first value, followed by the %%% equivalent of the standard UNIX wc (word %%% count) utility output of lines, words, and %%% characters. This is produced by Robert %%% Solovay's checksum utility.", %%% } %%% ==================================================================== @Preamble{"\input bibnames.sty"} %%% ==================================================================== %%% Acknowledgement abbreviations: @String{ack-nhfb= "Nelson H. F. Beebe, University of Utah, Department of Mathematics, 110 LCB, 155 S 1400 E RM 233, Salt Lake City, UT 84112-0090, USA, Tel: +1 801 581 5254, FAX: +1 801 581 4148, e-mail: \path|beebe@math.utah.edu|, \path|beebe@acm.org|, \path|beebe@computer.org| (Internet), URL: \path|http://www.math.utah.edu/~beebe/|"} %%% ==================================================================== %%% Journal abbreviations: @String{j-TOCT= "ACM Transactions on Computation Theory"} %%% ==================================================================== %%% Publisher abbreviations: @String{pub-ACM= "ACM Press"} @String{pub-ACM:adr= "New York, NY 10036, USA"} %%% ==================================================================== %%% Bibliography entries: @Article{Fortnow:2009:EF, author = "Lance Fortnow", title = "{Editor}'s Foreword", journal = j-TOCT, volume = "1", number = "1", pages = "1:1--1:??", month = feb, year = "2009", CODEN = "????", DOI = "http://doi.acm.org/10.1145/1490270.1490271", ISSN = "1942-3454 (print), 1942-3462 (electronic)", bibdate = "Fri Apr 24 19:03:42 MDT 2009", bibsource = "http://www.acm.org/pubs/contents/journals/toct/; http://www.math.utah.edu/pub/tex/bib/toct.bib", acknowledgement = ack-nhfb, articleno = "1", fjournal = "ACM Transactions on Computation Theory", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J1190", } @Article{Aaronson:2009:ANB, author = "Scott Aaronson and Avi Wigderson", title = "Algebrization: {A} New Barrier in Complexity Theory", journal = j-TOCT, volume = "1", number = "1", pages = "2:1--2:??", month = feb, year = "2009", CODEN = "????", DOI = "http://doi.acm.org/10.1145/1490270.1490272", ISSN = "1942-3454 (print), 1942-3462 (electronic)", bibdate = "Fri Apr 24 19:03:42 MDT 2009", bibsource = "http://www.acm.org/pubs/contents/journals/toct/; http://www.math.utah.edu/pub/tex/bib/toct.bib", abstract = "Any proof of P $\neq$ NP will have to overcome two barriers: {\em relativization\/} and {\em natural proofs}. Yet over the last decade, we have seen circuit lower bounds (e.g., that PP does not have linear-size circuits) that overcome both barriers simultaneously. So the question arises of whether there is a third barrier to progress on the central questions in complexity theory.\par In this article, we present such a barrier, which we call {\em algebraic relativization\/} or {\em algebrization}. The idea is that, when we relativize some complexity class inclusion, we should give the simulating machine access not only to an oracle $A$, but also to a low-degree extension of $A$ over a finite field or ring.\par We systematically go through basic results and open problems in complexity theory to delineate the power of the new algebrization barrier. First, we show that all known nonrelativizing results based on arithmetization---both inclusions such as IP = PSPACE and MIP = NEXP, and separations such as MA$_{{EXP}_n}$ P/poly---do indeed algebrize. Second, we show that almost all of the major open problems---including P versus NP, P versus RP, and NEXP versus P/poly---will require {\em non-algebrizing techniques}. In some cases, algebrization seems to explain exactly why progress stopped where it did: for example, why we have superlinear circuit lower bounds for PromiseMA but not for NP.\par Our second set of results follows from lower bounds in a new model of {\em algebraic query complexity}, which we introduce in this article and which is interesting in its own right. Some of our lower bounds use direct combinatorial and algebraic arguments, while others stem from a surprising connection between our model and communication complexity. Using this connection, we are also able to give an MA-protocol for the Inner Product function with $O(\sqrt{n} \log n)$ communication (essentially matching a lower bound of Klauck), as well as a communication complexity conjecture whose truth would imply NL $\neq$ NP.", acknowledgement = ack-nhfb, articleno = "2", fjournal = "ACM Transactions on Computation Theory", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J1190", keywords = "arithmetization; communication complexity; interactive proofs; Low-degree polynomials; oracles; query complexity", } @Article{Razborov:2009:SPB, author = "Alexander Razborov", title = "A Simple Proof of {Bazzi's Theorem}", journal = j-TOCT, volume = "1", number = "1", pages = "3:1--3:??", month = feb, year = "2009", CODEN = "????", DOI = "http://doi.acm.org/10.1145/1490270.1490273", ISSN = "1942-3454 (print), 1942-3462 (electronic)", bibdate = "Fri Apr 24 19:03:42 MDT 2009", bibsource = "http://www.acm.org/pubs/contents/journals/toct/; http://www.math.utah.edu/pub/tex/bib/toct.bib", abstract = "Linial and Nisan [1990] asked if any polylog-wise independent distribution fools any function in AC$^0$. In a recent remarkable development, Bazzi solved this problem for the case of DNF formulas. The aim of this note is to present a simplified version of his proof.", acknowledgement = ack-nhfb, articleno = "3", fjournal = "ACM Transactions on Computation Theory", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J1190", keywords = "DNF; Pseudo-random generators", } @Article{Bourke:2009:DPR, author = "Chris Bourke and Raghunath Tewari and N. V. Vinodchandran", title = "Directed Planar Reachability Is in Unambiguous Log-Space", journal = j-TOCT, volume = "1", number = "1", pages = "4:1--4:??", month = feb, year = "2009", CODEN = "????", DOI = "http://doi.acm.org/10.1145/1490270.1490274", ISSN = "1942-3454 (print), 1942-3462 (electronic)", bibdate = "Fri Apr 24 19:03:42 MDT 2009", bibsource = "http://www.acm.org/pubs/contents/journals/toct/; http://www.math.utah.edu/pub/tex/bib/toct.bib", abstract = "We make progress in understanding the complexity of the graph reachability problem in the context of unambiguous logarithmic space computation; a restricted form of nondeterminism. As our main result, we show a new upper bound on the {\em directed planar reachability problem\/} by showing that it can be decided in the class unambiguous logarithmic space (UL). We explore the possibility of showing the same upper bound for the general graph reachability problem. We give a simple reduction showing that the reachability problem for directed graphs with thickness two is complete for the class nondeterministic logarithmic space (NL). Hence an extension of our results to directed graphs with thickness two will unconditionally collapse NL to UL.", acknowledgement = ack-nhfb, articleno = "4", fjournal = "ACM Transactions on Computation Theory", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J1190", keywords = "Directed graph reachability; planar graphs; unambiguous log-space", } @Article{David:2009:ISB, author = "Matei David and Toniann Pitassi and Emanuele Viola", title = "Improved Separations between Nondeterministic and Randomized Multiparty Communication", journal = j-TOCT, volume = "1", number = "2", pages = "5:1--5:??", month = sep, year = "2009", CODEN = "????", DOI = "http://doi.acm.org/10.1145/1595391.1595392", ISSN = "1942-3454 (print), 1942-3462 (electronic)", bibdate = "Tue Mar 16 10:08:03 MDT 2010", bibsource = "http://www.acm.org/pubs/contents/journals/toct/; http://www.math.utah.edu/pub/tex/bib/toct.bib", acknowledgement = ack-nhfb, articleno = "5", fjournal = "ACM Transactions on Computation Theory", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J1190", } @Article{Guruswami:2009:HSS, author = "Venkatesan Guruswami and Prasad Raghavendra", title = "Hardness of Solving Sparse Overdetermined Linear Systems: {A} 3-Query {PCP} over Integers", journal = j-TOCT, volume = "1", number = "2", pages = "6:1--6:??", month = sep, year = "2009", CODEN = "????", DOI = "http://doi.acm.org/10.1145/1595391.1595393", ISSN = "1942-3454 (print), 1942-3462 (electronic)", bibdate = "Tue Mar 16 10:08:03 MDT 2010", bibsource = "http://www.acm.org/pubs/contents/journals/toct/; http://www.math.utah.edu/pub/tex/bib/toct.bib", acknowledgement = ack-nhfb, articleno = "6", fjournal = "ACM Transactions on Computation Theory", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J1190", } @Article{Ben-Sasson:2009:SQP, author = "Eli Ben-Sasson and Prahladh Harsha and Oded Lachish and Arie Matsliah", title = "Sound 3-Query {PCPPs} Are Long", journal = j-TOCT, volume = "1", number = "2", pages = "7:1--7:??", month = sep, year = "2009", CODEN = "????", DOI = "http://doi.acm.org/10.1145/1595391.1595394", ISSN = "1942-3454 (print), 1942-3462 (electronic)", bibdate = "Tue Mar 16 10:08:03 MDT 2010", bibsource = "http://www.acm.org/pubs/contents/journals/toct/; http://www.math.utah.edu/pub/tex/bib/toct.bib", acknowledgement = ack-nhfb, articleno = "7", fjournal = "ACM Transactions on Computation Theory", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J1190", } @Article{Kyncl:2010:LRD, author = "Jan Kyn{\v{c}}l and Tom{\'a}{\v{s}} Vysko{\v{c}}il", title = "Logspace Reduction of Directed Reachability for Bounded Genus Graphs to the Planar Case", journal = j-TOCT, volume = "1", number = "3", pages = "8:1--8:??", month = mar, year = "2010", CODEN = "????", DOI = "http://doi.acm.org/10.1145/1714450.1714451", ISSN = "1942-3454 (print), 1942-3462 (electronic)", bibdate = "Tue Mar 16 10:08:04 MDT 2010", bibsource = "http://www.acm.org/pubs/contents/journals/toct/; http://www.math.utah.edu/pub/tex/bib/toct.bib", acknowledgement = ack-nhfb, articleno = "8", fjournal = "ACM Transactions on Computation Theory", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J1190", } @Article{Beame:2010:FCD, author = "Paul Beame and Russell Impagliazzo and Toniann Pitassi and Nathan Segerlind", title = "Formula Caching in {DPLL}", journal = j-TOCT, volume = "1", number = "3", pages = "9:1--9:??", month = mar, year = "2010", CODEN = "????", DOI = "http://doi.acm.org/10.1145/1714450.1714452", ISSN = "1942-3454 (print), 1942-3462 (electronic)", bibdate = "Tue Mar 16 10:08:04 MDT 2010", bibsource = "http://www.acm.org/pubs/contents/journals/toct/; http://www.math.utah.edu/pub/tex/bib/toct.bib", acknowledgement = ack-nhfb, articleno = "9", fjournal = "ACM Transactions on Computation Theory", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J1190", } @Article{Datta:2010:PDP, author = "Samir Datta and Raghav Kulkarni and Nutan Limaye and Meena Mahajan", title = "Planarity, Determinants, Permanents, and (Unique) Matchings", journal = j-TOCT, volume = "1", number = "3", pages = "10:1--10:??", month = mar, year = "2010", CODEN = "????", DOI = "http://doi.acm.org/10.1145/1714450.1714453", ISSN = "1942-3454 (print), 1942-3462 (electronic)", bibdate = "Tue Mar 16 10:08:04 MDT 2010", bibsource = "http://www.acm.org/pubs/contents/journals/toct/; http://www.math.utah.edu/pub/tex/bib/toct.bib", acknowledgement = ack-nhfb, articleno = "10", fjournal = "ACM Transactions on Computation Theory", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J1190", } @Article{Yin:2010:CPP, author = "Yitong Yin", title = "Cell-Probe Proofs", journal = j-TOCT, volume = "2", number = "1", pages = "1:1--1:??", month = nov, year = "2010", CODEN = "????", DOI = "http://dx.doi.org/10.1145/1867719.1867720", ISSN = "1942-3454 (print), 1942-3462 (electronic)", bibdate = "Tue Nov 23 11:20:48 MST 2010", bibsource = "http://www.acm.org/pubs/contents/journals/toct/; http://www.math.utah.edu/pub/tex/bib/toct.bib", acknowledgement = ack-nhfb, articleno = "1", fjournal = "ACM Transactions on Computation Theory", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J1190", } @Article{Hoefer:2010:TAC, author = "Martin Hoefer and Alexander Souza", title = "Tradeoffs and Average-Case Equilibria in Selfish Routing", journal = j-TOCT, volume = "2", number = "1", pages = "2:1--2:??", month = nov, year = "2010", CODEN = "????", DOI = "http://dx.doi.org/10.1145/1867719.1867721", ISSN = "1942-3454 (print), 1942-3462 (electronic)", bibdate = "Tue Nov 23 11:20:48 MST 2010", bibsource = "http://www.acm.org/pubs/contents/journals/toct/; http://www.math.utah.edu/pub/tex/bib/toct.bib", acknowledgement = ack-nhfb, articleno = "2", fjournal = "ACM Transactions on Computation Theory", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J1190", } @Article{Purdy:2011:LBC, author = "Eric Purdy", title = "Lower Bounds for Coin-Weighing Problems", journal = j-TOCT, volume = "2", number = "2", pages = "3:1--3:??", month = mar, year = "2011", CODEN = "????", DOI = "http://dx.doi.org/10.1145/1944857.1944858", ISSN = "1942-3454 (print), 1942-3462 (electronic)", bibdate = "Mon Mar 28 09:50:20 MDT 2011", bibsource = "http://www.acm.org/pubs/contents/journals/toct/; http://www.math.utah.edu/pub/tex/bib/toct.bib", acknowledgement = ack-nhfb, articleno = "3", fjournal = "ACM Transactions on Computation Theory", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J1190", } @Article{Arvind:2011:SGI, author = "Vikraman Arvind and Jacobo Tor{\'a}n", title = "Solvable Group Isomorphism Is (Almost) in {NP} $\cap$ {coNP}", journal = j-TOCT, volume = "2", number = "2", pages = "4:1--4:??", month = mar, year = "2011", CODEN = "????", DOI = "http://dx.doi.org/10.1145/1944857.1944859", ISSN = "1942-3454 (print), 1942-3462 (electronic)", bibdate = "Mon Mar 28 09:50:20 MDT 2011", bibsource = "http://www.acm.org/pubs/contents/journals/toct/; http://www.math.utah.edu/pub/tex/bib/toct.bib", acknowledgement = ack-nhfb, articleno = "4", fjournal = "ACM Transactions on Computation Theory", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J1190", } @Article{Fellows:2011:CDF, author = "Michael R. Fellows and Jiong Guo and Hannes Moser and Rolf Niedermeier", title = "A Complexity Dichotomy for Finding Disjoint Solutions of Vertex Deletion Problems", journal = j-TOCT, volume = "2", number = "2", pages = "5:1--5:??", month = mar, year = "2011", CODEN = "????", DOI = "http://dx.doi.org/10.1145/1944857.1944860", ISSN = "1942-3454 (print), 1942-3462 (electronic)", bibdate = "Mon Mar 28 09:50:20 MDT 2011", bibsource = "http://www.acm.org/pubs/contents/journals/toct/; http://www.math.utah.edu/pub/tex/bib/toct.bib", acknowledgement = ack-nhfb, articleno = "5", fjournal = "ACM Transactions on Computation Theory", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J1190", } @Article{Hitchcock:2011:KCR, author = "John M. Hitchcock and A. Pavan and N. V. Vinodchandran", title = "{Kolmogorov} Complexity in Randomness Extraction", journal = j-TOCT, volume = "3", number = "1", pages = "1:1--1:??", month = aug, year = "2011", CODEN = "????", DOI = "http://dx.doi.org/10.1145/2003685.2003686", ISSN = "1942-3454 (print), 1942-3462 (electronic)", bibdate = "Mon Sep 5 16:58:07 MDT 2011", bibsource = "http://www.acm.org/pubs/contents/journals/toct/; http://www.math.utah.edu/pub/tex/bib/toct.bib", acknowledgement = ack-nhfb, articleno = "1", fjournal = "ACM Transactions on Computation Theory", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J1190", } @Article{Kulkarni:2011:PIP, author = "Raghav Kulkarni", title = "On the Power of Isolation in Planar Graphs", journal = j-TOCT, volume = "3", number = "1", pages = "2:1--2:??", month = aug, year = "2011", CODEN = "????", DOI = "http://dx.doi.org/10.1145/2003685.2003687", ISSN = "1942-3454 (print), 1942-3462 (electronic)", bibdate = "Mon Sep 5 16:58:07 MDT 2011", bibsource = "http://www.acm.org/pubs/contents/journals/toct/; http://www.math.utah.edu/pub/tex/bib/toct.bib", acknowledgement = ack-nhfb, articleno = "2", fjournal = "ACM Transactions on Computation Theory", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J1190", } @Article{Smyth:2011:AQC, author = "Clifford Smyth", title = "Approximate Query Complexity", journal = j-TOCT, volume = "3", number = "1", pages = "3:1--3:??", month = aug, year = "2011", CODEN = "????", DOI = "http://dx.doi.org/10.1145/2003685.2003688", ISSN = "1942-3454 (print), 1942-3462 (electronic)", bibdate = "Mon Sep 5 16:58:07 MDT 2011", bibsource = "http://www.acm.org/pubs/contents/journals/toct/; http://www.math.utah.edu/pub/tex/bib/toct.bib", acknowledgement = ack-nhfb, articleno = "3", fjournal = "ACM Transactions on Computation Theory", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J1190", } @Article{Cook:2012:PBP, author = "Stephen Cook and Pierre McKenzie and Dustin Wehr and Mark Braverman and Rahul Santhanam", title = "Pebbles and Branching Programs for Tree Evaluation", journal = j-TOCT, volume = "3", number = "2", pages = "4:1--4:??", month = jan, year = "2012", CODEN = "????", DOI = "http://dx.doi.org/10.1145/2077336.2077337", ISSN = "1942-3454 (print), 1942-3462 (electronic)", ISSN-L = "1942-3454", bibdate = "Fri Mar 16 15:22:48 MDT 2012", bibsource = "http://www.acm.org/pubs/contents/journals/toct/; http://www.math.utah.edu/pub/tex/bib/toct.bib", abstract = "We introduce the tree evaluation problem, show that it is in LogDCFL (and hence in $P$), and study its branching program complexity in the hope of eventually proving a superlogarithmic space lower bound. The input to the problem is a rooted, balanced $d$-ary tree of height h, whose internal nodes are labeled with $d$-ary functions on $[ k ] = {1,\ldots{} , k}$, and whose leaves are labeled with elements of $[ k ]$. Each node obtains a value in $[ k ]$ equal to its $d$-ary function applied to the values of its $d$ children. The output is the value of the root. We show that the standard black pebbling algorithm applied to the binary tree of height h yields a deterministic $k$-way branching program with $O(k h)$ states solving this problem, and we prove that this upper bound is tight for $h = 2$ and $h = 3$. We introduce a simple semantic restriction called thrifty on $k$-way branching programs solving tree evaluation problems and show that the same state bound of $\Theta ( k h)$ is tight for all $h \geq 2$ for deterministic thrifty programs. We introduce fractional pebbling for trees and show that this yields nondeterministic thrifty programs with $\Theta(k h/2 + 1)$ states solving the Boolean problem `determine whether the root has value 1', and prove that this bound is tight for $h = 2, 3, 4$. We also prove that this same bound is tight for unrestricted nondeterministic $k$-way branching programs solving the Boolean problem for $h = 2, 3$.", acknowledgement = ack-nhfb, articleno = "4", fjournal = "ACM Transactions on Computation Theory", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J1190", } @Article{Gal:2012:TQL, author = "Anna Gal and Andrew Mills", title = "Three-Query Locally Decodable Codes with Higher Correctness Require Exponential Length", journal = j-TOCT, volume = "3", number = "2", pages = "5:1--5:??", month = jan, year = "2012", CODEN = "????", DOI = "http://dx.doi.org/10.1145/2077336.2077338", ISSN = "1942-3454 (print), 1942-3462 (electronic)", ISSN-L = "1942-3454", bibdate = "Fri Mar 16 15:22:48 MDT 2012", bibsource = "http://www.acm.org/pubs/contents/journals/toct/; http://www.math.utah.edu/pub/tex/bib/toct.bib", abstract = "Locally decodable codes are error-correcting codes with the extra property that, in order to retrieve the value of a single input position, it is sufficient to read a small number of positions of the codeword. We refer to the probability of getting the correct value as the correctness of the decoding algorithm. A breakthrough result by Yekhanin [2007] showed that 3-query linear locally decodable codes may have subexponential length. The construction of Yekhanin, and the three query constructions that followed, achieve correctness only up to a certain limit which is 1--3 $\delta$ for nonbinary codes, where an adversary is allowed to corrupt up to $\delta$ fraction of the codeword. The largest correctness for a subexponential length 3-query binary code is achieved in a construction by Woodruff [2008], and it is below 1--3 $\delta$. We show that achieving slightly larger correctness (as a function of $\delta$) requires exponential codeword length for 3-query codes. Previously, there were no larger than quadratic lower bounds known for locally decodable codes with more than 2 queries, even in the case of 3-query linear codes. Our lower bounds hold for linear codes over arbitrary finite fields and for binary nonlinear codes. Considering larger number of queries, we obtain lower bounds for $q$-query codes for $q > 3$, under certain assumptions on the decoding algorithm that have been commonly used in previous constructions. We also prove bounds on the largest correctness achievable by these decoding algorithms, regardless of the length of the code. Our results explain the limitations on correctness in previous constructions using such decoding algorithms. In addition, our results imply trade-offs on the parameters of error-correcting data structures.", acknowledgement = ack-nhfb, articleno = "5", fjournal = "ACM Transactions on Computation Theory", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J1190", } @Article{Beame:2012:VMR, author = "Paul Beame and Trinh Huynh", title = "The Value of Multiple {Read\slash} Write Streams for Approximating Frequency Moments", journal = j-TOCT, volume = "3", number = "2", pages = "6:1--6:??", month = jan, year = "2012", CODEN = "????", DOI = "http://dx.doi.org/10.1145/2077336.2077339", ISSN = "1942-3454 (print), 1942-3462 (electronic)", ISSN-L = "1942-3454", bibdate = "Fri Mar 16 15:22:48 MDT 2012", bibsource = "http://www.acm.org/pubs/contents/journals/toct/; http://www.math.utah.edu/pub/tex/bib/toct.bib", abstract = "We consider the read/write streams model, an extension of the standard data stream model in which an algorithm can create and manipulate multiple read/write streams in addition to its input data stream. Like the data stream model, the most important parameter for this model is the amount of internal memory used by such an algorithm. The other key parameters are the number of streams the algorithm uses and the number of passes it makes on these streams. We consider how the addition of multiple streams impacts the ability of algorithms to approximate the frequency moments of the input stream. We show that any randomized read/write stream algorithm with a fixed number of streams and a sublogarithmic number of passes that produces a constant factor approximation of the $k$ -th frequency moment $F_k$ of an input sequence of length of at most $N$ from $\{1,\ldots{},N\}$ requires space $\Omega(N^{1 - 4/k - \delta})$ for any $\delta > 0$. For comparison, it is known that with a single read-only one-pass data stream there is a randomized constant-factor approximation for $F_k$ using $\tilde{O}(N^{1 - 2/k})$ space, and that by sorting, which can be done deterministically in $O(\log N)$ passes using $3$ read/write streams, $F_k$ can be computed exactly. Therefore, although the ability to manipulate multiple read/write streams can add substantial power to the data stream model, with a sublogarithmic number of passes this does not significantly improve the ability to approximate higher frequency moments efficiently. We obtain our results by showing a new connection between the read/write streams model and the multiparty number-in-hand communication model.", acknowledgement = ack-nhfb, articleno = "6", fjournal = "ACM Transactions on Computation Theory", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J1190", } @Article{Tani:2012:EQA, author = "Seiichiro Tani and Hirotada Kobayashi and Keiji Matsumoto", title = "Exact Quantum Algorithms for the Leader Election Problem", journal = j-TOCT, volume = "4", number = "1", pages = "1:1--1:??", month = mar, year = "2012", CODEN = "????", DOI = "http://dx.doi.org/10.1145/2141938.2141939", ISSN = "1942-3454 (print), 1942-3462 (electronic)", ISSN-L = "1942-3454", bibdate = "Tue Nov 6 18:23:48 MST 2012", bibsource = "http://www.acm.org/pubs/contents/journals/toct/; http://www.math.utah.edu/pub/tex/bib/toct.bib", abstract = "This article gives a separation between quantum and classical models in pure (i.e., noncryptographic) computing abilities with no restriction on the amount of available computing resources, by considering the exact solvability of the leader election problem in anonymous networks, a celebrated unsolvable problem in classical distributed computing. The goal of the leader election problem is to elect a unique leader from among distributed parties. In an anonymous network, all parties with the same number of communication links are identical. It is well-known that no classical algorithm can exactly solve (i.e., in bounded time without error) the leader election problem in anonymous networks, even if the number of parties is given. This article devises a quantum algorithm that, if the number of parties is given, exactly solves the problem for any network topology in polynomial rounds with polynomial communication/time complexity with respect to the number of parties, when the parties are connected with quantum communication links and they have the ability of quantum computing. Our algorithm works even when only an upper bound of the number of parties is given. In such a case, no classical algorithm can solve the problem even under the zero-error setting, the setting in which error is not allowed but running time may be unbounded.", acknowledgement = ack-nhfb, articleno = "1", fjournal = "ACM Transactions on Computation Theory", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J1190", } @Article{Cheraghchi:2012:ALT, author = "Mahdi Cheraghchi and Johan H{\aa}stad and Marcus Isaksson and Ola Svensson", title = "Approximating Linear Threshold Predicates", journal = j-TOCT, volume = "4", number = "1", pages = "2:1--2:??", month = mar, year = "2012", CODEN = "????", DOI = "http://dx.doi.org/10.1145/2141938.2141940", ISSN = "1942-3454 (print), 1942-3462 (electronic)", ISSN-L = "1942-3454", bibdate = "Tue Nov 6 18:23:48 MST 2012", bibsource = "http://www.acm.org/pubs/contents/journals/toct/; http://www.math.utah.edu/pub/tex/bib/toct.bib", abstract = "We study constraint satisfaction problems on the domain {-1, 1}, where the given constraints are homogeneous linear threshold predicates, that is, predicates of the form $\sgn(w_1 x_1 + \cdots + w_n x_n)$ for some positive integer weights $w_1, \ldots{}, w_n$. Despite their simplicity, current techniques fall short of providing a classification of these predicates in terms of approximability. In fact, it is not easy to guess whether there exists a homogeneous linear threshold predicate that is approximation resistant or not. The focus of this article is to identify and study the approximation curve of a class of threshold predicates that allow for nontrivial approximation. Arguably the simplest such predicate is the majority predicate $\sgn(x_1 + \cdots + x_n)$, for which we obtain an almost complete understanding of the asymptotic approximation curve, assuming the Unique Games Conjecture. Our techniques extend to a more general class of ``majority-like'' predicates and we obtain parallel results for them. In order to classify these predicates, we introduce the notion of Chow-robustness that might be of independent interest.", acknowledgement = ack-nhfb, articleno = "2", fjournal = "ACM Transactions on Computation Theory", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J1190", } @Article{De:2012:ELB, author = "Anindya De and Thomas Watson", title = "Extractors and Lower Bounds for Locally Samplable Sources", journal = j-TOCT, volume = "4", number = "1", pages = "3:1--3:??", month = mar, year = "2012", CODEN = "????", DOI = "http://dx.doi.org/10.1145/2141938.2141941", ISSN = "1942-3454 (print), 1942-3462 (electronic)", ISSN-L = "1942-3454", bibdate = "Tue Nov 6 18:23:48 MST 2012", bibsource = "http://www.acm.org/pubs/contents/journals/toct/; http://www.math.utah.edu/pub/tex/bib/toct.bib", abstract = "We consider the problem of extracting randomness from sources that are efficiently samplable, in the sense that each output bit of the sampler only depends on some small number $d$ of the random input bits. As our main result, we construct a deterministic extractor that, given any $d$-local source with min-entropy $k$ on $n$ bits, extracts $\Omega(k^2 / n d)$ bits that are $2^{-n \Omega (1)}$-close to uniform, provided $d \leq o(\log n)$ and $k \geq n^{2/3 + \gamma}$ (for arbitrarily small constants $\gamma > 0$). Using our result, we also improve a result of Viola [2010] who proved a $1/2 O(1/\log n)$ statistical distance lower bound for $o(\log n)$-local samplers trying to sample input-output pairs of an explicit boolean function, assuming the samplers use at most $n + n^{1 - \delta}$ random bits for some constant $\delta > 0$. Using a different function, we simultaneously improve the lower bound to $1/2 - 2^{-n \Omega (1)}$ and eliminate the restriction on the number of random bits.", acknowledgement = ack-nhfb, articleno = "3", fjournal = "ACM Transactions on Computation Theory", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J1190", } @Article{Schoenebeck:2012:CCN, author = "Grant R. Schoenebeck and Salil Vadhan", title = "The Computational Complexity of {Nash} Equilibria in Concisely Represented Games", journal = j-TOCT, volume = "4", number = "2", pages = "4:1--4:??", month = may, year = "2012", CODEN = "????", DOI = "http://dx.doi.org/10.1145/2189778.2189779", ISSN = "1942-3454 (print), 1942-3462 (electronic)", ISSN-L = "1942-3454", bibdate = "Tue Nov 6 18:23:49 MST 2012", bibsource = "http://www.acm.org/pubs/contents/journals/toct/; http://www.math.utah.edu/pub/tex/bib/toct.bib", abstract = "Games may be represented in many different ways, and different representations of games affect the complexity of problems associated with games, such as finding a Nash equilibrium. The traditional method of representing a game is to explicitly list all the payoffs, but this incurs an exponential blowup as the number of agents grows. We study two models of concisely represented games: circuit games, where the payoffs are computed by a given boolean circuit, and graph games, where each agent's payoff is a function of only the strategies played by its neighbors in a given graph. For these two models, we study the complexity of four questions: determining if a given strategy is a Nash equilibrium, finding a Nash equilibrium, determining if there exists a pure Nash equilibrium, and determining if there exists a Nash equilibrium in which the payoffs to a player meet some given guarantees. In many cases, we obtain tight results, showing that the problems are complete for various complexity classes.", acknowledgement = ack-nhfb, articleno = "4", fjournal = "ACM Transactions on Computation Theory", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J1190", } @Article{Kawamura:2012:CTO, author = "Akitoshi Kawamura and Stephen Cook", title = "Complexity Theory for Operators in Analysis", journal = j-TOCT, volume = "4", number = "2", pages = "5:1--5:??", month = may, year = "2012", CODEN = "????", DOI = "http://dx.doi.org/10.1145/2189778.2189780", ISSN = "1942-3454 (print), 1942-3462 (electronic)", ISSN-L = "1942-3454", bibdate = "Tue Nov 6 18:23:49 MST 2012", bibsource = "http://www.acm.org/pubs/contents/journals/toct/; http://www.math.utah.edu/pub/tex/bib/toct.bib", abstract = "We propose an extension of the framework for discussing the computational complexity of problems involving uncountably many objects, such as real numbers, sets and functions, that can be represented only through approximation. The key idea is to use a certain class of string functions as names representing these objects. These are more expressive than infinite sequences, which served as names in prior work that formulated complexity in more restricted settings. An advantage of using string functions is that we can define their size in a way inspired by higher-type complexity theory. This enables us to talk about computation on string functions whose time or space is bounded polynomially in the input size, giving rise to more general analogues of the classes P, NP, and PSPACE. We also define NP- and PSPACE-completeness under suitable many-one reductions. Because our framework separates machine computation and semantics, it can be applied to problems on sets of interest in analysis once we specify a suitable representation (encoding). As prototype applications, we consider the complexity of functions (operators) on real numbers, real sets, and real functions. For example, the task of numerical algorithms for solving a certain class of differential equations is naturally viewed as an operator taking real functions to real functions. As there was no complexity theory for operators, previous results only stated how complex the solution can be. We now reformulate them and show that the operator itself is polynomial-space complete.", acknowledgement = ack-nhfb, articleno = "5", fjournal = "ACM Transactions on Computation Theory", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J1190", } @Article{Penna:2012:CRM, author = "Paolo Penna and Carmine Ventre", title = "Collusion-Resistant Mechanisms with Verification Yielding Optimal Solutions", journal = j-TOCT, volume = "4", number = "2", pages = "6:1--6:??", month = may, year = "2012", CODEN = "????", DOI = "http://dx.doi.org/10.1145/2189778.2189781", ISSN = "1942-3454 (print), 1942-3462 (electronic)", ISSN-L = "1942-3454", bibdate = "Tue Nov 6 18:23:49 MST 2012", bibsource = "http://www.acm.org/pubs/contents/journals/toct/; http://www.math.utah.edu/pub/tex/bib/toct.bib", abstract = "A truthful mechanism consists of an algorithm augmented with a suitable payment function that guarantees that the players cannot improve their utilities by cheating. Mechanism design approaches are particularly appealing for designing protocols that cannot be manipulated by rational players. We present new constructions of so-called mechanisms with verification introduced by Nisan and Ronen [2001]. We first show how to obtain mechanisms that, for single-parameter domains, are resistant to coalitions of colluding agents even if they can exchange compensations. Based on this result we derive a class of exact truthful mechanisms with verification for arbitrary bounded domains. This class of problems includes most of the problems studied in the algorithmic mechanism design literature and for which exact solutions cannot be obtained with truthful mechanisms without verification. This result is an improvement over all known previous constructions of exact mechanisms with verification.", acknowledgement = ack-nhfb, articleno = "6", fjournal = "ACM Transactions on Computation Theory", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J1190", } @Article{Beyersdorff:2012:PBD, author = "Olaf Beyersdorff and Nicola Galesi and Massimo Lauria and Alexander A. Razborov", title = "Parameterized Bounded-Depth {Frege} Is not Optimal", journal = j-TOCT, volume = "4", number = "3", pages = "7:1--7:??", month = sep, year = "2012", CODEN = "????", DOI = "http://dx.doi.org/10.1145/2355580.2355582", ISSN = "1942-3454 (print), 1942-3462 (electronic)", ISSN-L = "1942-3454", bibdate = "Tue Nov 6 18:23:50 MST 2012", bibsource = "http://www.acm.org/pubs/contents/journals/toct/; http://www.math.utah.edu/pub/tex/bib/toct.bib", abstract = "A general framework for parameterized proof complexity was introduced by Dantchev et al. [2007]. There, the authors show important results on tree-like Parameterized Resolution---a parameterized version of classical Resolution---and their gap complexity theorem implies lower bounds for that system. The main result of this article significantly improves upon this by showing optimal lower bounds for a parameterized version of bounded-depth Frege. More precisely, we prove that the pigeonhole principle requires proofs of size n$^{\Omega (k)}$ in parameterized bounded-depth Frege, and, as a special case, in dag-like Parameterized Resolution. This answers an open question posed in Dantchev et al. [2007]. In the opposite direction, we interpret a well-known technique for FPT algorithms as a DPLL procedure for Parameterized Resolution. Its generalization leads to a proof search algorithm for Parameterized Resolution that in particular shows that tree-like Parameterized Resolution allows short refutations of all parameterized contradictions given as bounded-width CNFs.", acknowledgement = ack-nhfb, articleno = "7", fjournal = "ACM Transactions on Computation Theory", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J1190", } @Article{Watson:2012:RWW, author = "Thomas Watson", title = "Relativized Worlds without Worst-Case to Average-Case Reductions for {NP}", journal = j-TOCT, volume = "4", number = "3", pages = "8:1--8:??", month = sep, year = "2012", CODEN = "????", DOI = "http://dx.doi.org/10.1145/2355580.2355583", ISSN = "1942-3454 (print), 1942-3462 (electronic)", ISSN-L = "1942-3454", bibdate = "Tue Nov 6 18:23:50 MST 2012", bibsource = "http://www.acm.org/pubs/contents/journals/toct/; http://www.math.utah.edu/pub/tex/bib/toct.bib", abstract = "We prove that, relative to an oracle, there is no worst-case to average-case reduction for NP. We also handle classes that are somewhat larger than NP, as well as worst-case to errorless -average-case reductions. In fact, we prove that relative to an oracle, there is no worst-case to errorless-average-case reduction from NP to BPP$_{||}^{NP}$. We also handle reductions from NP to the polynomial-time hierarchy and beyond, under strong restrictions on the number of queries the reductions can make.", acknowledgement = ack-nhfb, articleno = "8", fjournal = "ACM Transactions on Computation Theory", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J1190", } @Article{Kayal:2012:SSR, author = "Neeraj Kayal and Chandan Saha", title = "On the Sum of Square Roots of Polynomials and Related Problems", journal = j-TOCT, volume = "4", number = "4", pages = "9:1--9:??", month = nov, year = "2012", CODEN = "????", DOI = "http://dx.doi.org/10.1145/2382559.2382560", ISSN = "1942-3454 (print), 1942-3462 (electronic)", ISSN-L = "1942-3454", bibdate = "Sun May 5 09:31:28 MDT 2013", bibsource = "http://www.acm.org/pubs/contents/journals/toct/; http://www.math.utah.edu/pub/tex/bib/toct.bib", abstract = "The sum of square roots over integers problem is the task of deciding the sign of a nonzero sum, {$ S = \Sigma_{i = 1}^n \delta_i \cdot \sqrt a_i $}, where \delta $_i$ \in {+1, -1} and a$_i$ 's are positive integers that are upper bounded by {$N$} (say). A fundamental open question in numerical analysis and computational geometry is whether {$ | S | \geq 1 / 2^{(n \cdot \log N) O(1)} $} when {$ S \neq 0 $}. We study a formulation of this problem over polynomials. Given an expression {$ S = \Sigma_{i = 1}^n c_i \cdot \sqrt f_i (x) $}, where $ c_i $'s belong to a field of characteristic $0$ and $ f_i $'s are univariate polynomials with degree bounded by $d$ and $ f_i(0) \neq 0 $ for all $i$, is it true that the minimum exponent of $x$ that has a nonzero coefficient in the power series {$S$} is upper bounded by {$ (n \cdot d)^{O(1)} $}, unless {$ S = 0 $}? We answer this question affirmatively. Further, we show that this result over polynomials can be used to settle (positively) the sum of square roots problem for a special class of integers: Suppose each integer $ a_i $ is of the form, {$ a_i = X^d_i + b_{i1} X^{di - 1} + \ldots {} + b_{idi} $}, $ d_i > 0 $, where {$X$} is a positive real number and $ b_{ij} $'s are integers. Let {$ B = \max (| b_{ij} |_{i, j}, 1) $} and $ d = \max_i \{ d_i \} $. If {$ X > (B + 1)^{(n \cdot d)O(1)} $} then a nonzero {$ S = \Sigma_{i = 1}^n \delta_i \sqrt a_i $} is lower bounded as {$ | S | \geq 1 / X^(n \cdot d)O(1) $}. The constant in {$ O (1) $}, as fixed by our analysis, is roughly $2$. We then consider the following more general problem. Given an arithmetic circuit computing a multivariate polynomial {$ f (X) $} and integer $d$, is the degree of {$ f (X) $} less than or equal to $d$ ? We give a {coRP$^{PP}$}-algorithm for this problem, improving previous results of Allender et al. [2009] and Koiran and Perifel [2007].", acknowledgement = ack-nhfb, articleno = "9", fjournal = "ACM Transactions on Computation Theory", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J1190", } @Article{Pass:2012:PRT, author = "Rafael Pass and Muthuramakrishnan Venkitasubramaniam", title = "A Parallel Repetition Theorem for Constant-Round {Arthur--Merlin} Proofs", journal = j-TOCT, volume = "4", number = "4", pages = "10:1--10:??", month = nov, year = "2012", CODEN = "????", DOI = "http://dx.doi.org/10.1145/2382559.2382561", ISSN = "1942-3454 (print), 1942-3462 (electronic)", ISSN-L = "1942-3454", bibdate = "Sun May 5 09:31:28 MDT 2013", bibsource = "http://www.acm.org/pubs/contents/journals/toct/; http://www.math.utah.edu/pub/tex/bib/toct.bib", abstract = "We show a parallel-repetition theorem for constant-round Arthur--Merlin Proofs, using an efficient reduction. As a consequence, we show that parallel repetition reduces the soundness-error at an optimal rate (up to a negligible factor) in constant-round public-coin argument systems, and constant-round public-coin proofs of knowledge. The first of these results resolves an open question posed by Bellare, Impagliazzo, and Naor (FOCS'97).", acknowledgement = ack-nhfb, articleno = "10", fjournal = "ACM Transactions on Computation Theory", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J1190", } @Article{Ron:2012:AIM, author = "Dana Ron and Ronitt Rubinfeld and Muli Safra and Alex Samorodnitsky and Omri Weinstein", title = "Approximating the Influence of Monotone {Boolean} Functions in {$ O(\sqrt n) $} Query Complexity", journal = j-TOCT, volume = "4", number = "4", pages = "11:1--11:??", month = nov, year = "2012", CODEN = "????", DOI = "http://dx.doi.org/10.1145/2382559.2382562", ISSN = "1942-3454 (print), 1942-3462 (electronic)", ISSN-L = "1942-3454", bibdate = "Sun May 5 09:31:28 MDT 2013", bibsource = "http://www.acm.org/pubs/contents/journals/toct/; http://www.math.utah.edu/pub/tex/bib/toct.bib", abstract = "The Total Influence ( Average Sensitivity ) of a discrete function is one of its fundamental measures. We study the problem of approximating the total influence of a monotone Boolean function, which we denote by {$ I[f] $}. We present a randomized algorithm that approximates the influence of such functions to within a multiplicative factor of $ (1 \pm \epsilon) $ by performing {$ O(\sqrt n I[f] \poly (1 / \epsilon)) $} queries. We also prove a lower bound of {$ \Omega (\sqrt n \log n \cdot I[f]) $} on the query complexity of any constant factor approximation algorithm for this problem (which holds for {$ I[f] = \Omega (1) $}), hence showing that our algorithm is almost optimal in terms of its dependence on $n$. For general functions, we give a lower bound of {$ \Omega (n I[f]) $}, which matches the complexity of a simple sampling algorithm.", acknowledgement = ack-nhfb, articleno = "11", fjournal = "ACM Transactions on Computation Theory", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J1190", } @Article{Vlassis:2012:CCS, author = "Nikos Vlassis and Michael L. Littman and David Barber", title = "On the Computational Complexity of Stochastic Controller Optimization in {POMDPs}", journal = j-TOCT, volume = "4", number = "4", pages = "12:1--12:??", month = nov, year = "2012", CODEN = "????", DOI = "http://dx.doi.org/10.1145/2382559.2382563", ISSN = "1942-3454 (print), 1942-3462 (electronic)", ISSN-L = "1942-3454", bibdate = "Sun May 5 09:31:28 MDT 2013", bibsource = "http://www.acm.org/pubs/contents/journals/toct/; http://www.math.utah.edu/pub/tex/bib/toct.bib", abstract = "We show that the problem of finding an optimal stochastic blind controller in a Markov decision process is an NP-hard problem. The corresponding decision problem is NP-hard in PSPACE and sqrt-sum-hard, hence placing it in NP would imply breakthroughs in long-standing open problems in computer science. Our result establishes that the more general problem of stochastic controller optimization in POMDPs is also NP-hard. Nonetheless, we outline a special case that is convex and admits efficient global solutions.", acknowledgement = ack-nhfb, articleno = "12", fjournal = "ACM Transactions on Computation Theory", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J1190", } @Article{Austrin:2013:UP, author = "Per Austrin and Johan H{\aa}stad", title = "On the usefulness of predicates", journal = j-TOCT, volume = "5", number = "1", pages = "1:1--1:??", month = may, year = "2013", CODEN = "????", ISSN = "1942-3454 (print), 1942-3462 (electronic)", ISSN-L = "1942-3454", bibdate = "Thu Dec 12 17:32:04 MST 2013", bibsource = "http://www.acm.org/pubs/contents/journals/toct/; http://www.math.utah.edu/pub/tex/bib/toct.bib", abstract = "Motivated by the pervasiveness of strong inapproximability results for Max-CSPs, we introduce a relaxed notion of an approximate solution of a Max-CSP. In this relaxed version, loosely speaking, the algorithm is allowed to replace the constraints of an instance by some other (possibly real-valued) constraints, and then only needs to satisfy as many of the new constraints as possible. To be more precise, we introduce the following notion of a predicate $P$ being useful for a (real-valued) objective $Q$: given an almost satisfiable Max- $P$ instance, there is an algorithm that beats a random assignment on the corresponding Max-$Q$ instance applied to the same sets of literals. The standard notion of a nontrivial approximation algorithm for a Max-CSP with predicate $P$ is exactly the same as saying that $P$ is useful for $P$ itself. We say that $P$ is useless if it is not useful for any $Q$. This turns out to be equivalent to the following pseudo-randomness property: given an almost satisfiable instance of Max- $P$, it is hard to find an assignment such that the induced distribution on k -bit strings defined by the instance is not essentially uniform. Under the unique games conjecture, we give a complete and simple characterization of useful Max-CSPs defined by a predicate: such a Max-CSP is useless if and only if there is a pairwise independent distribution supported on the satisfying assignments of the predicate. It is natural to also consider the case when no negations are allowed in the CSP instance, and we derive a similar complete characterization (under the UGC) there as well. Finally, we also include some results and examples shedding additional light on the approximability of certain Max-CSPs.", acknowledgement = ack-nhfb, articleno = "1", fjournal = "ACM Transactions on Computation Theory", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J1190", } @Article{Beyersdorff:2013:VPC, author = "Olaf Beyersdorff and Samir Datta and Andreas Krebs and Meena Mahajan and Gido Scharfenberger-Fabian and Karteek Sreenivasaiah and Michael Thomas and Heribert Vollmer", title = "Verifying proofs in constant depth", journal = j-TOCT, volume = "5", number = "1", pages = "2:1--2:??", month = may, year = "2013", CODEN = "????", ISSN = "1942-3454 (print), 1942-3462 (electronic)", ISSN-L = "1942-3454", bibdate = "Thu Dec 12 17:32:04 MST 2013", bibsource = "http://www.acm.org/pubs/contents/journals/toct/; http://www.math.utah.edu/pub/tex/bib/toct.bib", abstract = "In this paper we initiate the study of proof systems where verification of proofs proceeds by NC$^0$ circuits. We investigate the question which languages admit proof systems in this very restricted model. Formulated alternatively, we ask which languages can be enumerated by NC$^0$ functions. Our results show that the answer to this problem is not determined by the complexity of the language. On the one hand, we construct NC$^0$ proof systems for a variety of languages ranging from regular to NP complete. On the other hand, we show by combinatorial methods that even easy regular languages such as Exact-OR do not admit NC$^0$ proof systems. We also show that Majority does not admit NC$^0$ proof systems. Finally, we present a general construction of NC$^0$ proof systems for regular languages with strongly connected NFA's.", acknowledgement = ack-nhfb, articleno = "2", fjournal = "ACM Transactions on Computation Theory", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J1190", } @Article{Cygan:2013:MCP, author = "Marek Cygan and Marcin Pilipczuk and Michal Pilipczuk and Jakub Onufry Wojtaszczyk", title = "On multiway cut parameterized above lower bounds", journal = j-TOCT, volume = "5", number = "1", pages = "3:1--3:??", month = may, year = "2013", CODEN = "????", ISSN = "1942-3454 (print), 1942-3462 (electronic)", ISSN-L = "1942-3454", bibdate = "Thu Dec 12 17:32:04 MST 2013", bibsource = "http://www.acm.org/pubs/contents/journals/toct/; http://www.math.utah.edu/pub/tex/bib/toct.bib", abstract = "We introduce a concept of parameterizing a problem above the optimum solution of its natural linear programming relaxation and prove that the node multiway cut problem is fixed-parameter tractable (FPT) in this setting. As a consequence we prove that node multiway cut is FPT, when parameterized above the maximum separating cut, resolving an open problem of Razgon. Our results imply $ O^*(4^k) $ algorithms for vertex cover above maximum matching and almost 2-SAT as well as an $ O^*(2^k) $ algorithm for node multiway cut with a standard parameterization by the solution size, improving previous bounds for these problems.", acknowledgement = ack-nhfb, articleno = "3", fjournal = "ACM Transactions on Computation Theory", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J1190", } @Article{Englert:2013:EC, author = "Matthias Englert and Heiko R{\"o}glin and Jacob Sp{\"o}nemann and Berthold V{\"o}cking", title = "Economical Caching", journal = j-TOCT, volume = "5", number = "2", pages = "4:1--4:??", month = jul, year = "2013", CODEN = "????", DOI = "http://dx.doi.org/10.1145/2493246.2493247", ISSN = "1942-3454 (print), 1942-3462 (electronic)", ISSN-L = "1942-3454", bibdate = "Thu Dec 12 17:32:08 MST 2013", bibsource = "http://www.acm.org/pubs/contents/journals/toct/; http://www.math.utah.edu/pub/tex/bib/toct.bib", abstract = "We study the management of buffers and storages in environments with unpredictably varying prices in a competitive analysis. In the economical caching problem, there is a storage with a certain capacity. For each time step, an online algorithm is given a price from the interval $ [1, \alpha] $, a consumption, and possibly a buying limit. The online algorithm has to decide the amount to purchase from some commodity, knowing the parameter $ \alpha $ but without knowing how the price evolves in the future. The algorithm can purchase at most the buying limit. If it purchases more than the current consumption, then the excess is stored in the storage; otherwise, the gap between consumption and purchase must be taken from the storage. The goal is to minimize the total cost. Interesting motivating applications are, for example, stream caching on mobile devices with different classes of service, battery management in micro hybrid cars, and the efficient purchase of resources. First we consider the simple but natural class of algorithms that can informally be described as memoryless. We show that these algorithms cannot achieve a competitive ratio below $ \sqrt \alpha $. Then we present a more sophisticated deterministic algorithm achieving a competitive ratio of where $W$ denotes the Lambert $W$ function. We prove that this algorithm is optimal and that not even randomized online algorithms can achieve a better competitive ratio. On the other hand, we show how to achieve a constant competitive ratio if the storage capacity of the online algorithm exceeds the storage capacity of an optimal offline algorithm by a factor of $ \log \alpha $.", acknowledgement = ack-nhfb, articleno = "4", fjournal = "ACM Transactions on Computation Theory", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J1190", } @Article{Bogdanov:2013:HFL, author = "Andrej Bogdanov and Akinori Kawachi and Hidetoki Tanaka", title = "Hard Functions for Low-Degree Polynomials over Prime Fields", journal = j-TOCT, volume = "5", number = "2", pages = "5:1--5:??", month = jul, year = "2013", CODEN = "????", DOI = "http://dx.doi.org/10.1145/2493246.2493248", ISSN = "1942-3454 (print), 1942-3462 (electronic)", ISSN-L = "1942-3454", bibdate = "Thu Dec 12 17:32:08 MST 2013", bibsource = "http://www.acm.org/pubs/contents/journals/toct/; http://www.math.utah.edu/pub/tex/bib/toct.bib", abstract = "In this article, we present a new hardness amplification for low-degree polynomials over prime fields, namely, we prove that if some function is mildly hard to approximate by any low-degree polynomials then the sum of independent copies of the function is very hard to approximate by them. This result generalizes the XOR lemma for low-degree polynomials over the binary field given by Viola and Wigderson [2008]. The main technical contribution is the analysis of the Gowers norm over prime fields. For the analysis, we discuss a generalized low-degree test, which we call the Gowers test, for polynomials over prime fields, which is a natural generalization of that over the binary field given by Alon et al. [2003]. This Gowers test provides a new technique to analyze the Gowers norm over prime fields. Actually, the rejection probability of the Gowers test can be analyzed in the framework of Kaufman and Sudan [2008]. However, our analysis is self-contained and quantitatively better. By using our argument, we also prove the hardness of modulo functions for low-degree polynomials over prime fields.", acknowledgement = ack-nhfb, articleno = "5", fjournal = "ACM Transactions on Computation Theory", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J1190", } @Article{Williams:2013:ATP, author = "Ryan Williams", title = "Alternation-Trading Proofs, Linear Programming, and Lower Bounds", journal = j-TOCT, volume = "5", number = "2", pages = "6:1--6:??", month = jul, year = "2013", CODEN = "????", DOI = "http://dx.doi.org/10.1145/2493246.2493249", ISSN = "1942-3454 (print), 1942-3462 (electronic)", ISSN-L = "1942-3454", bibdate = "Thu Dec 12 17:32:08 MST 2013", bibsource = "http://www.acm.org/pubs/contents/journals/toct/; http://www.math.utah.edu/pub/tex/bib/toct.bib", abstract = "A fertile area of recent research has demonstrated concrete polynomial-time lower bounds for natural hard problems on restricted computational models. Among these problems are Satisfiability, Vertex Cover, Hamilton Path, MOD$_6$-SAT, Majority-of-Majority-SAT, and Tautologies, to name a few. The proofs of these lower bounds follow a proof-by-contradiction strategy that we call resource trading or alternation trading. An important open problem is to determine how powerful such proofs can possibly be. We propose a methodology for studying these proofs that makes them amenable to both formal analysis and automated theorem proving. We prove that the search for better lower bounds can often be turned into a problem of solving a large series of linear programming instances. Implementing a small-scale theorem prover based on these results, we extract new human-readable time lower bounds for several problems and identify patterns that allow for further generalization. The framework can also be used to prove concrete limitations on the current techniques.", acknowledgement = ack-nhfb, articleno = "6", fjournal = "ACM Transactions on Computation Theory", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J1190", } @Article{Ron:2013:ANR, author = "Dana Ron and Gilad Tsur", title = "On Approximating the Number of Relevant Variables in a Function", journal = j-TOCT, volume = "5", number = "2", pages = "7:1--7:??", month = jul, year = "2013", CODEN = "????", DOI = "http://dx.doi.org/10.1145/2493246.2493250", ISSN = "1942-3454 (print), 1942-3462 (electronic)", ISSN-L = "1942-3454", bibdate = "Thu Dec 12 17:32:08 MST 2013", bibsource = "http://www.acm.org/pubs/contents/journals/toct/; http://www.math.utah.edu/pub/tex/bib/toct.bib", abstract = "In this work we consider the problem of approximating the number of relevant variables in a function given query access to the function. Since obtaining a multiplicative factor approximation is hard in general, we consider several relaxations of the problem. In particular, we consider a relaxation of the property testing variant of the problem and we consider relaxations in which we have a promise that the function belongs to a certain family of functions (e.g., linear functions). In the former relaxation the task is to distinguish between the case that the number of relevant variables is at most $k$, and the case in which it is far from any function in which the number of relevant variables is more than $ (1 + \gamma) k $ for a parameter $ \gamma $. We give both upper bounds and almost matching lower bounds for the relaxations we study.", acknowledgement = ack-nhfb, articleno = "7", fjournal = "ACM Transactions on Computation Theory", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J1190", } @Article{Allender:2013:ISI, author = "Eric Allender and Shafi Goldwasser", title = "Introduction to the special issue on innovations in theoretical computer science 2012", journal = j-TOCT, volume = "5", number = "3", pages = "8:1--8:??", month = aug, year = "2013", DOI = "http://dx.doi.org/10.1145/2493252.2493253", ISSN = "1942-3454 (print), 1942-3462 (electronic)", ISSN-L = "1942-3454", bibdate = "Thu Dec 12 17:32:12 MST 2013", bibsource = "http://www.acm.org/pubs/contents/journals/toct/; http://www.math.utah.edu/pub/tex/bib/toct.bib", note = "Special issue on innovations in theoretical computer science 2012.", acknowledgement = ack-nhfb, articleno = "8", fjournal = "ACM Transactions on Computation Theory", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J1190", } @Article{Pagh:2013:CMM, author = "Rasmus Pagh", title = "Compressed matrix multiplication", journal = j-TOCT, volume = "5", number = "3", pages = "9:1--9:??", month = aug, year = "2013", DOI = "http://dx.doi.org/10.1145/2493252.2493254", ISSN = "1942-3454 (print), 1942-3462 (electronic)", ISSN-L = "1942-3454", bibdate = "Thu Dec 12 17:32:12 MST 2013", bibsource = "http://www.acm.org/pubs/contents/journals/toct/; http://www.math.utah.edu/pub/tex/bib/toct.bib", note = "Special issue on innovations in theoretical computer science 2012.", abstract = "We present a simple algorithm that approximates the product of $n$-by-$n$ real matrices $A$ and $B$. Let $ || A B ||_F $ denote the Frobenius norm of $ A B $, and $b$ be a parameter determining the time\slash accuracy trade-off. Given $2$-wise independent hash functions $ h_1, h_2 : [n] \to [b] $, and $ s_1, s_2 : [n] \to \{ - 1, + 1 \} $ the algorithm works by first ``compressing'' the matrix product into the polynomial $ p (x) = \Sigma_{k = 1}^n \left (\Sigma_{i = 1}^n A_{ik} s_1 (i) x^{h 1 (i)} \right) \left (\Sigma_{j = 1}^n B_{kj} s_2 (j) x^{h 2 (j)} \right) $. Using the fast Fourier transform to compute polynomial multiplication, we can compute $ c_0, \ldots {}, c_{b - 1} $ such that $ \Sigma_i c_i x^i = (p (x) \bmod x^b) + (p (x) \div x^b) $ in time $ {\~ O}(n^2 + n b) $. An unbiased estimator of $ (A B)_{ij} $ with variance at most $ || A B ||_F^2 / b $ can then be computed as: $ C_{ij} = s_1 (i) s_2 (j) c_{(h_1 (i) + h_2 (j)) \bmod b} $. Our approach also leads to an algorithm for computing AB exactly, with high probability, in time $ {\~ O}(N + n b) $ in the case where $A$ and $B$ have at most $N$ nonzero entries, and $ A B $ has at most $b$ nonzero entries.", acknowledgement = ack-nhfb, articleno = "9", fjournal = "ACM Transactions on Computation Theory", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J1190", } @Article{Viderman:2013:LTD, author = "Michael Viderman", title = "Linear-time decoding of regular expander codes", journal = j-TOCT, volume = "5", number = "3", pages = "10:1--10:??", month = aug, year = "2013", DOI = "http://dx.doi.org/10.1145/2493252.2493255", ISSN = "1942-3454 (print), 1942-3462 (electronic)", ISSN-L = "1942-3454", bibdate = "Thu Dec 12 17:32:12 MST 2013", bibsource = "http://www.acm.org/pubs/contents/journals/toct/; http://www.math.utah.edu/pub/tex/bib/toct.bib", note = "Special issue on innovations in theoretical computer science 2012.", abstract = "Sipser and Spielman (IEEE IT, [1996]) showed that any $ (c, d) $-regular expander code with expansion parameter $ > 3 / 4 $ is decodable in linear time from a constant fraction of errors. Feldman et al. (IEEE IT, [2007]) proved that expansion parameter $ > 2 / 3 + (1 / 3) c $ is sufficient to correct a constant fraction of errors in polynomial time using LP decoding. In this work, we give a simple combinatorial algorithm that achieves even better parameters. In particular, our algorithm runs in linear time and works for any expansion parameter $ > 2 / 3 - (1 / 6) c $. We also prove that our decoding algorithm can be executed in logarithmic time on a linear number of parallel processors.", acknowledgement = ack-nhfb, articleno = "10", fjournal = "ACM Transactions on Computation Theory", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J1190", } @Article{Ozols:2013:QRS, author = "Maris Ozols and Martin Roetteler and J{\'e}r{\'e}mie Roland", title = "Quantum rejection sampling", journal = j-TOCT, volume = "5", number = "3", pages = "11:1--11:??", month = aug, year = "2013", DOI = "http://dx.doi.org/10.1145/2493252.2493256", ISSN = "1942-3454 (print), 1942-3462 (electronic)", ISSN-L = "1942-3454", bibdate = "Thu Dec 12 17:32:12 MST 2013", bibsource = "http://www.acm.org/pubs/contents/journals/toct/; http://www.math.utah.edu/pub/tex/bib/toct.bib", note = "Special issue on innovations in theoretical computer science 2012.", abstract = "Rejection sampling is a well-known method to sample from a target distribution, given the ability to sample from a given distribution. The method has been first formalized by von Neumann [1951] and has many applications in classical computing. We define a quantum analogue of rejection sampling: given a black box producing a coherent superposition of (possibly unknown) quantum states with some amplitudes, the problem is to prepare a coherent superposition of the same states, albeit with different target amplitudes. The main result of this article is a tight characterization of the query complexity of this quantum state generation problem. We exhibit an algorithm, which we call quantum rejection sampling, and analyze its cost using semidefinite programming. Our proof of a matching lower bound is based on the automorphism principle that allows to symmetrize any algorithm over the automorphism group of the problem. Our main technical innovation is an extension of the automorphism principle to continuous groups that arise for quantum state generation problems where the oracle encodes unknown quantum states, instead of just classical data. Furthermore, we illustrate how quantum rejection sampling may be used as a primitive in designing quantum algorithms, by providing three different applications. We first show that it was implicitly used in the quantum algorithm for linear systems of equations by Harrow et al. [2009]. Second we show that it can be used to speed up the main step in the quantum Metropolis sampling algorithm by Temme et al. [2011]. Finally, we derive a new quantum algorithm for the hidden shift problem of an arbitrary Boolean function and relate its query complexity to ``water-filling'' of the Fourier spectrum.", acknowledgement = ack-nhfb, articleno = "11", fjournal = "ACM Transactions on Computation Theory", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J1190", } @Article{Drucker:2013:HCP, author = "Andrew Drucker", title = "High-confidence predictions under adversarial uncertainty", journal = j-TOCT, volume = "5", number = "3", pages = "12:1--12:??", month = aug, year = "2013", DOI = "http://dx.doi.org/10.1145/2493252.2493257", ISSN = "1942-3454 (print), 1942-3462 (electronic)", ISSN-L = "1942-3454", bibdate = "Thu Dec 12 17:32:12 MST 2013", bibsource = "http://www.acm.org/pubs/contents/journals/toct/; http://www.math.utah.edu/pub/tex/bib/toct.bib", note = "Special issue on innovations in theoretical computer science 2012.", abstract = "We study the setting in which the bits of an unknown infinite binary sequence $x$ are revealed sequentially to an observer. We show that very limited assumptions about $x$ allow one to make successful predictions about unseen bits of $x$. First, we study the problem of successfully predicting a single 0 from among the bits of $x$. In our model, we have only one chance to make a prediction, but may do so at a time of our choosing. This model is applicable to a variety of situations in which we want to perform an action of fixed duration, and need to predict a ``safe'' time-interval to perform it. Letting $ N_t $ denote the number of $1$'s among the first $t$ bits of $x$, we say that $x$ is ``$ \epsilon $-weakly sparse'' if $ \lim \inf (N_t / t) < = \epsilon $. Our main result is a randomized algorithm that, given any $ \epsilon $-weakly sparse sequence $x$, predicts a $0$ of $x$ with success probability as close as desired to $ 1 - \epsilon $. Thus, we can perform this task with essentially the same success probability as under the much stronger assumption that each bit of $x$ takes the value $1$ independently with probability $ \epsilon $. We apply this result to show how to successfully predict a bit ($0$ or $1$ ) under a broad class of possible assumptions on the sequence $x$. The assumptions are stated in terms of the behavior of a finite automaton $M$ reading the bits of $x$. We also propose and solve a variant of the well-studied ``ignorant forecasting'' problem. For every $ \epsilon > 0 $, we give a randomized forecasting algorithm $ S_\epsilon $ that, given sequential access to a binary sequence $x$, makes a prediction of the form: ``A $p$ fraction of the next $N$ bits will be $1$'s.'' (The algorithm gets to choose $p$, $N$, and the time of the prediction.) For any fixed sequence $x$, the forecast fraction $p$ is accurate to within $ \pm {} \epsilon $ with probability $ 1 - \epsilon $.", acknowledgement = ack-nhfb, articleno = "12", fjournal = "ACM Transactions on Computation Theory", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J1190", } @Article{Chattopadhyay:2013:GIA, author = "Arkadev Chattopadhyay and Jacobo Tor{\'a}n and Fabian Wagner", title = "Graph Isomorphism is Not {AC$^0$}-Reducible to Group Isomorphism", journal = j-TOCT, volume = "5", number = "4", pages = "13:1--13:??", month = nov, year = "2013", CODEN = "????", DOI = "http://dx.doi.org/10.1145/2540088", ISSN = "1942-3454 (print), 1942-3462 (electronic)", ISSN-L = "1942-3454", bibdate = "Thu Dec 12 17:32:15 MST 2013", bibsource = "http://www.acm.org/pubs/contents/journals/toct/; http://www.math.utah.edu/pub/tex/bib/toct.bib", abstract = "We give a new upper bound for the Group and Quasigroup Isomorphism problems when the input structures are given explicitly by multiplication tables. We show that these problems can be computed by polynomial size nondeterministic circuits of unbounded fan-in with $ O(\log \log n) $ depth and $ O(\log^2 n) $ nondeterministic bits, where n is the number of group elements. This improves the existing upper bound for the problems. In the previous bound the circuits have bounded fan-in but depth $ O(\log^2 n) $ and also $ O(\log^2 n) $ nondeterministic bits. We then prove that the kind of circuits from our upper bound cannot compute the Parity function. Since Parity is AC$^0$-reducible to Graph Isomorphism, this implies that Graph Isomorphism is strictly harder than Group or Quasigroup Isomorphism under the ordering defined by AC$^0$ reductions. We extend this result to the stronger ACC$^0 [p]$ reduction and its randomized version.", acknowledgement = ack-nhfb, articleno = "13", fjournal = "ACM Transactions on Computation Theory", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J1190", } @Article{De:2013:EOH, author = "Anindya De and Elchanan Mossel", title = "Explicit Optimal Hardness via {Gaussian} Stability Results", journal = j-TOCT, volume = "5", number = "4", pages = "14:1--14:??", month = nov, year = "2013", CODEN = "????", DOI = "http://dx.doi.org/10.1145/2505766", ISSN = "1942-3454 (print), 1942-3462 (electronic)", ISSN-L = "1942-3454", bibdate = "Thu Dec 12 17:32:15 MST 2013", bibsource = "http://www.acm.org/pubs/contents/journals/toct/; http://www.math.utah.edu/pub/tex/bib/toct.bib", abstract = "The results of Raghavendra [2008] show that assuming Khot's Unique Games Conjecture [2002], for every constraint satisfaction problem there exists a generic semidefinite program that achieves the optimal approximation factor. This result is existential as it does not provide an explicit optimal rounding procedure nor does it allow to calculate exactly the Unique Games hardness of the problem. Obtaining an explicit optimal approximation scheme and the corresponding approximation factor is a difficult challenge for each specific approximation problem. Khot et al. [2004] established a general approach for determining the exact approximation factor and the corresponding optimal rounding algorithm for any given constraint satisfaction problem. However, this approach crucially relies on results explicitly proving optimal partitions in the Gaussian space. Until recently, Borell's result [1985] was the only nontrivial Gaussian partition result known. In this article we derive the first explicit optimal approximation algorithm and the corresponding approximation factor using a new result on Gaussian partitions due to Isaksson and Mossel [2012]. This Gaussian result allows us to determine the exact Unique Games Hardness of MAX-$3$-EQUAL. In particular, our results show that Zwick's algorithm for this problem achieves the optimal approximation factor and prove that the approximation achieved by the algorithm is $ \approx 0.796 $ as conjectured by Zwick [1998]. We further use the previously known optimal Gaussian partitions results to obtain a new Unique Games Hardness factor for MAX-$k$-CSP: Using the well-known fact that jointly normal pairwise independent random variables are fully independent, we show that the UGC hardness of Max-$k$-CSP is $ \lceil (k + 1) / 2 \rceil 2^{k - 1} $, improving on results of Austrin and Mossel [2009].", acknowledgement = ack-nhfb, articleno = "14", fjournal = "ACM Transactions on Computation Theory", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J1190", } @Article{Dalmau:2013:RSC, author = "V{\'\i}ctor Dalmau and Andrei Krokhin", title = "Robust Satisfiability for {CSPs}: Hardness and Algorithmic Results", journal = j-TOCT, volume = "5", number = "4", pages = "15:1--15:??", month = nov, year = "2013", CODEN = "????", DOI = "http://dx.doi.org/10.1145/2540090", ISSN = "1942-3454 (print), 1942-3462 (electronic)", ISSN-L = "1942-3454", bibdate = "Thu Dec 12 17:32:15 MST 2013", bibsource = "http://www.acm.org/pubs/contents/journals/toct/; http://www.math.utah.edu/pub/tex/bib/toct.bib", abstract = "An algorithm for a constraint satisfaction problem is called robust if it outputs an assignment satisfying at least a $ (1 - f (\epsilon)) $-fraction of constraints for each $ (1 - \epsilon) $-satisfiable instance (i.e., such that at most a \epsilon -fraction of constraints needs to be removed to make the instance satisfiable), where $ f(\epsilon) \to 0 $ as $ \epsilon \to 0 $. We establish an algebraic framework for analyzing constraint satisfaction problems admitting an efficient robust algorithm with functions $f$ of a given growth rate. We use this framework to derive hardness results. We also describe three classes of problems admitting an efficient robust algorithm such that $f$ is $ O (1 / \log (1 / \epsilon)) $, $ O(\epsilon^{1 / k}) $ for some $ k > 1 $, and $ O(\epsilon) $, respectively. Finally, we give a complete classification of robust satisfiability with a given $f$ for the Boolean case.", acknowledgement = ack-nhfb, articleno = "15", fjournal = "ACM Transactions on Computation Theory", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J1190", } @Article{Fellows:2013:DFP, author = "Michael Fellows and Fedor V. Fomin and Daniel Lokshtanov and Elena Losievskaja and Frances Rosamond and Saket Saurabh", title = "Distortion is Fixed Parameter Tractable", journal = j-TOCT, volume = "5", number = "4", pages = "16:1--16:??", month = nov, year = "2013", CODEN = "????", DOI = "http://dx.doi.org/10.1145/2489789", ISSN = "1942-3454 (print), 1942-3462 (electronic)", ISSN-L = "1942-3454", bibdate = "Thu Dec 12 17:32:15 MST 2013", bibsource = "http://www.acm.org/pubs/contents/journals/toct/; http://www.math.utah.edu/pub/tex/bib/toct.bib", abstract = "We study low-distortion embedding of metric spaces into the line, and more generally, into the shortest path metric of trees, from the parameterized complexity perspective. Let $ M = M (G) $ be the shortest path metric of an edge-weighted graph $G$, with the vertex set $ V (G) $ and the edge set $ E (G) $, on $n$ vertices. We give the first fixed parameter tractable algorithm that for an unweighted graph metric $M$ and integer $d$ either constructs an embedding of $M$ into the line with distortion at most $d$, or concludes that no such embedding exists. Our algorithm requires O( nd$^4$ (2 d + 1)$^{2d}$ ) time which is a significant improvement over the best previous algorithm that runs in time $ O(n^{4d + 2} d^{O(1)}) $. Because of its apparent similarity to the notoriously hard Bandwidth Minimization problem, we find it surprising that this problem turns out to be fixed parameter tractable. We extend our results on embedding unweighted graph metric into the line in two ways. First, we give an algorithm to construct small-distortion embeddings of weighted graph metrics. The running time of our algorithm is $ O(n (d W)^4 (2 d + 1)^{2dW}) $, where $W$ is the largest edge weight of the input graph. To complement this result, we show that the exponential dependence on the maximum edge weight is unavoidable. In particular, we show that deciding whether a weighted graph metric $ M (G) $ with maximum weight $ W < | V (G)| $ can be embedded into the line with distortion at most $d$ is NP-complete for every fixed rational $ d \geq 2 $. This rules out any possibility of an algorithm with running time $ O((n W)^{h(d)}) $ where $h$ is a function of $d$ alone. Second, we consider more general host metrics for which analogous results hold. In particular, we prove that for any tree $T$ with maximum degree \Delta , embedding $M$ into a shortest path metric of $T$ is fixed parameter tractable, parameterized by $ (\Delta, d) $.", acknowledgement = ack-nhfb, articleno = "16", fjournal = "ACM Transactions on Computation Theory", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J1190", } @Article{Razborov:2013:RA, author = "Alexander Razborov and Emanuele Viola", title = "Real Advantage", journal = j-TOCT, volume = "5", number = "4", pages = "17:1--17:??", month = nov, year = "2013", CODEN = "????", DOI = "http://dx.doi.org/10.1145/2540089", ISSN = "1942-3454 (print), 1942-3462 (electronic)", ISSN-L = "1942-3454", bibdate = "Thu Dec 12 17:32:15 MST 2013", bibsource = "http://www.acm.org/pubs/contents/journals/toct/; http://www.math.utah.edu/pub/tex/bib/toct.bib", abstract = "We highlight the challenge of proving correlation bounds between boolean functions and real-valued polynomials, where any non-boolean output counts against correlation. We prove that real-valued polynomials of degree $ 1 2 \lg_2 \lg_2 n $ have correlation with parity at most zero. Such a result is false for modular and threshold polynomials. Its proof is based on a variant of an anti-concentration result by Costello et al. [2006].", acknowledgement = ack-nhfb, articleno = "17", fjournal = "ACM Transactions on Computation Theory", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J1190", } @Article{Harkins:2013:ELA, author = "Ryan C. Harkins and John M. Hitchcock", title = "Exact Learning Algorithms, Betting Games, and Circuit Lower Bounds", journal = j-TOCT, volume = "5", number = "4", pages = "18:1--18:??", month = nov, year = "2013", CODEN = "????", DOI = "http://dx.doi.org/10.1145/2539126.2539130", ISSN = "1942-3454 (print), 1942-3462 (electronic)", ISSN-L = "1942-3454", bibdate = "Thu Dec 12 17:32:15 MST 2013", bibsource = "http://www.acm.org/pubs/contents/journals/toct/; http://www.math.utah.edu/pub/tex/bib/toct.bib", abstract = "This article extends and improves the work of Fortnow and Klivans [2009], who showed that if a circuit class $C$ has an efficient learning algorithm in Angluin's model of exact learning via equivalence and membership queries [Angluin 1988], then we have the lower bound EXP$^{NP}$ not $C$. We use entirely different techniques involving betting games [Buhrman et al. 2001] to remove the NP oracle and improve the lower bound to EXP not $C$. This shows that it is even more difficult to design a learning algorithm for $C$ than the results of Fortnow and Klivans [2009] indicated. We also investigate the connection between betting games and natural proofs, and as a corollary the existence of strong pseudorandom generators. Our results also yield further evidence that the class of Boolean circuits has no efficient exact learning algorithm. This is because our separation is strong in that it yields a natural proof [Razborov and Rudich 1997] against the class. From this we conclude that an exact learning algorithm for Boolean circuits would imply that strong pseudorandom generators do not exist, which contradicts widely believed conjectures from cryptography. As a corollary we obtain that if strong pseudorandom generators exist, then there is no exact learning algorithm for Boolean circuits.", acknowledgement = ack-nhfb, articleno = "18", fjournal = "ACM Transactions on Computation Theory", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J1190", } @Article{Ada:2014:HBP, author = "Anil Ada and Arkadev Chattopadhyay and Stephen A. Cook and Lila Fontes and Michal Kouck{\'y} and Toniann Pitassi", title = "The Hardness of Being Private", journal = j-TOCT, volume = "6", number = "1", pages = "1:1--1:??", month = mar, year = "2014", CODEN = "????", DOI = "http://dx.doi.org/10.1145/2567671", ISSN = "1942-3454 (print), 1942-3462 (electronic)", ISSN-L = "1942-3454", bibdate = "Tue Apr 1 06:02:31 MDT 2014", bibsource = "http://www.acm.org/pubs/contents/journals/toct/; http://www.math.utah.edu/pub/tex/bib/toct.bib", abstract = "Kushilevitz [1989] initiated the study of information-theoretic privacy within the context of communication complexity. Unfortunately, it has been shown that most interesting functions are not privately computable [Kushilevitz 1989, Brandt and Sandholm 2008]. The unattainability of perfect privacy for many functions motivated the study of approximate privacy. Feigenbaum et al. [2010a, 2010b] define notions of worst-case as well as average-case approximate privacy and present several interesting upper bounds as well as some open problems for further study. In this article, we obtain asymptotically tight bounds on the trade-offs between both the worst-case and average-case approximate privacy of protocols and their communication cost for Vickrey auctions. Further, we relate the notion of average-case approximate privacy to other measures based on information cost of protocols. This enables us to prove exponential lower bounds on the subjective approximate privacy of protocols for computing the Intersection function, independent of its communication cost. This proves a conjecture of Feigenbaum et al. [2010a].", acknowledgement = ack-nhfb, articleno = "1", fjournal = "ACM Transactions on Computation Theory", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J1190", } @Article{Austrin:2014:NNH, author = "Per Austrin and Ryan O'Donnell and Li-Yang Tan and John Wright", title = "New {NP}-Hardness Results for $3$-Coloring and $2$-to-$1$ Label Cover", journal = j-TOCT, volume = "6", number = "1", pages = "2:1--2:??", month = mar, year = "2014", CODEN = "????", DOI = "http://dx.doi.org/10.1145/2537800", ISSN = "1942-3454 (print), 1942-3462 (electronic)", ISSN-L = "1942-3454", bibdate = "Tue Apr 1 06:02:31 MDT 2014", bibsource = "http://www.acm.org/pubs/contents/journals/toct/; http://www.math.utah.edu/pub/tex/bib/toct.bib", abstract = "We show that given a 3-colorable graph, it is NP-hard to find a 3-coloring with $ (16 / 17 + \epsilon) $ of the edges bichromatic. In a related result, we show that given a satisfiable instance of the 2-to-1 Label Cover problem, it is NP-hard to find a $ (23 / 24 + \epsilon) $-satisfying assignment.", acknowledgement = ack-nhfb, articleno = "2", fjournal = "ACM Transactions on Computation Theory", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J1190", } @Article{Glasser:2014:UDN, author = "Christian Gla{\ss}er and John M. Hitchcock and A. Pavan and Stephan Travers", title = "Unions of Disjoint {NP}-Complete Sets", journal = j-TOCT, volume = "6", number = "1", pages = "3:1--3:??", month = mar, year = "2014", CODEN = "????", DOI = "http://dx.doi.org/10.1145/2591508", ISSN = "1942-3454 (print), 1942-3462 (electronic)", ISSN-L = "1942-3454", bibdate = "Tue Apr 1 06:02:31 MDT 2014", bibsource = "http://www.acm.org/pubs/contents/journals/toct/; http://www.math.utah.edu/pub/tex/bib/toct.bib", abstract = "We study the following question: if A and B are disjoint NP-complete sets, then is A \cup B NP-complete? We provide necessary and sufficient conditions under which the union of disjoint NP-complete sets remain complete.", acknowledgement = ack-nhfb, articleno = "3", fjournal = "ACM Transactions on Computation Theory", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J1190", } @Article{Sun:2014:ECN, author = "Shu-Ming Sun and Ning Zhong", title = "On Effective Convergence of Numerical Solutions for Differential Equations", journal = j-TOCT, volume = "6", number = "1", pages = "4:1--4:??", month = mar, year = "2014", CODEN = "????", DOI = "http://dx.doi.org/10.1145/2578219", ISSN = "1942-3454 (print), 1942-3462 (electronic)", ISSN-L = "1942-3454", bibdate = "Tue Apr 1 06:02:31 MDT 2014", bibsource = "http://www.acm.org/pubs/contents/journals/toct/; http://www.math.utah.edu/pub/tex/bib/toct.bib", abstract = "This article studies the effective convergence of numerical solutions of initial value problems (IVPs) for ordinary differential equations (ODEs). A convergent sequence $ \{ Y_m \} $ of numerical solutions is said to be effectively convergent to the exact solution if there is an algorithm that computes an $ N \in N $, given an arbitrary $ n \in N $ as input, such that the error between $ Y_m $ and the exact solution is less than $ 2^{-n} $ for all $ m \geq N $. It is proved that there are convergent numerical solutions generated from Euler's method which are not effectively convergent. It is also shown that a theoretically-proved-computable solution using Picard's iteration method might not be computable by classical numerical methods, which suggests that sometimes there is a gap between theoretical computability and practical numerical computations concerning solutions of ODEs. Moreover, it is noted that the main theorem (Theorem 4.1) provides an example of an IVP with a nonuniform Lipschitz function for which the numerical solutions generated by Euler's method are still convergent.", acknowledgement = ack-nhfb, articleno = "4", fjournal = "ACM Transactions on Computation Theory", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J1190", } @Article{ODonnell:2014:OLB, author = "Ryan O'Donnell and Yi Wu and Yuan Zhou", title = "Optimal Lower Bounds for Locality-Sensitive Hashing (Except When $q$ is Tiny)", journal = j-TOCT, volume = "6", number = "1", pages = "5:1--5:??", month = mar, year = "2014", CODEN = "????", DOI = "http://dx.doi.org/10.1145/2578221", ISSN = "1942-3454 (print), 1942-3462 (electronic)", ISSN-L = "1942-3454", bibdate = "Tue Apr 1 06:02:31 MDT 2014", bibsource = "http://www.acm.org/pubs/contents/journals/toct/; http://www.math.utah.edu/pub/tex/bib/hash.bib; http://www.math.utah.edu/pub/tex/bib/toct.bib", abstract = "We study lower bounds for Locality-Sensitive Hashing (LSH) in the strongest setting: point sets in $ \{ 0, 1 \}^d $ under the Hamming distance. Recall that $H$ is said to be an $ (r, c r, p, q) $-sensitive hash family if all pairs $ x, y \in \{ 0, 1 \}^d $ with $ {\rm dist}(x, y) \leq r $ have probability at least $p$ of collision under a randomly chosen $ h \in H $, whereas all pairs $ x, y \in \{ 0, 1 \}^d $ with $ {\rm dist}(x, y) \geq c r $ have probability at most $q$ of collision. Typically, one considers $ d \to \infty $, with $ c > 1 $ fixed and $q$ bounded away from $0$. For its applications to approximate nearest-neighbor search in high dimensions, the quality of an LSH family $H$ is governed by how small its $ \rho $ parameter $ \rho = \ln (1 / p) / l n(1 / q) $ is as a function of the parameter $c$. The seminal paper of Indyk and Motwani [1998] showed that for each $ c \geq 1 $, the extremely simple family $ H = \{ x \mapsto x $ _i$ : i \in [d] \} $ achieves $ \rho \leq 1 / c $. The only known lower bound, due to Motwani et al. [2007], is that $ \rho $ must be at least $ (e^{1 / c} - 1) / (e^{1 / c} + 1) \geq .46 / c $ (minus $ o_d(1) $ ). The contribution of this article is twofold. (1) We show the ``optimal'' lower bound for $ \rho $: it must be at least $ 1 / c $ (minus $ o_d(1) $ ). Our proof is very simple, following almost immediately from the observation that the noise stability of a boolean function at time $t$ is a log-convex function of $t$. (2) We raise and discuss the following issue: neither the application of LSH to nearest-neighbor search nor the known LSH lower bounds hold as stated if the q parameter is tiny. Here, ``tiny'' means $ q = 2^{- \Theta (d)} $, a parameter range we believe is natural.", acknowledgement = ack-nhfb, articleno = "5", fjournal = "ACM Transactions on Computation Theory", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J1190", } @Article{Cygan:2014:CCG, author = "Marek Cygan and Stefan Kratsch and Marcin Pilipczuk and Michal Pilipczuk and Magnus Wahlstr{\"o}m", title = "Clique Cover and Graph Separation: New Incompressibility Results", journal = j-TOCT, volume = "6", number = "2", pages = "6:1--6:??", month = may, year = "2014", CODEN = "????", DOI = "http://dx.doi.org/10.1145/2594439", ISSN = "1942-3454 (print), 1942-3462 (electronic)", ISSN-L = "1942-3454", bibdate = "Thu Jun 5 14:42:53 MDT 2014", bibsource = "http://www.acm.org/pubs/contents/journals/toct/; http://www.math.utah.edu/pub/tex/bib/toct.bib", abstract = "The field of kernelization studies polynomial-time preprocessing routines for hard problems in the framework of parameterized complexity. In this article, we show that, unless the polynomial hierarchy collapses to its third level, the following parameterized problems do not admit a polynomial-time preprocessing algorithm that reduces the size of an instance to polynomial in the parameter: ---Edge Clique Cover, parameterized by the number of cliques, ---Directed Edge/Vertex Multiway Cut, parameterized by the size of the cutset, even in the case of two terminals, ---Edge/Vertex Multicut, parameterized by the size of the cutset, and --- k -Way Cut, parameterized by the size of the cutset.", acknowledgement = ack-nhfb, articleno = "6", fjournal = "ACM Transactions on Computation Theory", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J1190", } @Article{Chen:2014:HIA, author = "Yijia Chen and J{\"o}rg Flum and Moritz M{\"u}ller", title = "Hard Instances of Algorithms and Proof Systems", journal = j-TOCT, volume = "6", number = "2", pages = "7:1--7:??", month = may, year = "2014", CODEN = "????", DOI = "http://dx.doi.org/10.1145/2601336", ISSN = "1942-3454 (print), 1942-3462 (electronic)", ISSN-L = "1942-3454", bibdate = "Thu Jun 5 14:42:53 MDT 2014", bibsource = "http://www.acm.org/pubs/contents/journals/toct/; http://www.math.utah.edu/pub/tex/bib/toct.bib", abstract = "If the class TAUT of tautologies of propositional logic has no almost optimal algorithm, then every algorithm A deciding TAUT has a hard sequence, that is, a polynomial time computable sequence witnessing that A is not almost optimal. We show that this result extends to every $\Pi t p$-complete problem with $t \geq 1$; however, assuming the Measure Hypothesis, there is a problem which has no almost optimal algorithm but is decided by an algorithm without hard sequences. For problems Q with an almost optimal algorithm, we analyze whether every algorithm deciding Q, which is not almost optimal, has a hard sequence.", acknowledgement = ack-nhfb, articleno = "7", fjournal = "ACM Transactions on Computation Theory", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J1190", } @Article{Goldberg:2014:CAC, author = "Leslie Ann Goldberg and Mark Jerrum", title = "The Complexity of Approximately Counting Tree Homomorphisms", journal = j-TOCT, volume = "6", number = "2", pages = "8:1--8:??", month = may, year = "2014", CODEN = "????", DOI = "http://dx.doi.org/10.1145/2600917", ISSN = "1942-3454 (print), 1942-3462 (electronic)", ISSN-L = "1942-3454", bibdate = "Thu Jun 5 14:42:53 MDT 2014", bibsource = "http://www.acm.org/pubs/contents/journals/toct/; http://www.math.utah.edu/pub/tex/bib/toct.bib", abstract = "We study two computational problems, parameterised by a fixed tree H. \#HOMSTO( H ) is the problem of counting homomorphisms from an input graph G to H. \#WHOMSTO( H ) is the problem of counting weighted homomorphisms to H, given an input graph G and a weight function for each vertex v of G. Even though H is a tree, these problems turn out to be sufficiently rich to capture all of the known approximation behaviour in \# P. We give a complete trichotomy for \#WHOMSTO( H ). If H is a star, then \#WHOMSTO( H ) is in FP. If H is not a star but it does not contain a certain induced subgraph J 3, then \#WHOMSTO( H ) is equivalent under approximation-preserving (AP) reductions to \#BIS, the problem of counting independent sets in a bipartite graph. This problem is complete for the class \#RH \Pi 1 under AP-reductions. Finally, if H contains an induced J$_3$, then \#WHOMSTO( H ) is equivalent under AP-reductions to \#SAT, the problem of counting satisfying assignments to a CNF Boolean formula. Thus, \#WHOMSTO( H ) is complete for \#P under AP-reductions. The results are similar for \#HOMSTO( H ) except that a rich structure emerges if H contains an induced J$_3$. We show that there are trees H for which \#HOMSTO( H ) is \# SAT -equivalent (disproving a plausible conjecture of Kelk). However, it is still not known whether \#HOMSTO( H ) is \#SAT-hard for every tree H which contains an induced J 3. It turns out that there is an interesting connection between these homomorphism-counting problems and the problem of approximating the partition function of the ferromagnetic Potts model. In particular, we show that for a family of graphs Jq, parameterised by a positive integer q, the problem \#HOMSTO( Jq ) is AP-interreducible with the problem of approximating the partition function of the q -state Potts model. It was not previously known that the Potts model had a homomorphism-counting interpretation. We use this connection to obtain some additional upper bounds for the approximation complexity of \#HOMSTO( Jq ).", acknowledgement = ack-nhfb, articleno = "8", fjournal = "ACM Transactions on Computation Theory", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J1190", } @Article{Etessami:2014:NCC, author = "Kousha Etessami and Alistair Stewart and Mihalis Yannakakis", title = "A Note on the Complexity of Comparing Succinctly Represented Integers, with an Application to Maximum Probability Parsing", journal = j-TOCT, volume = "6", number = "2", pages = "9:1--9:??", month = may, year = "2014", CODEN = "????", DOI = "http://dx.doi.org/10.1145/2601327", ISSN = "1942-3454 (print), 1942-3462 (electronic)", ISSN-L = "1942-3454", bibdate = "Thu Jun 5 14:42:53 MDT 2014", bibsource = "http://www.acm.org/pubs/contents/journals/toct/; http://www.math.utah.edu/pub/tex/bib/toct.bib", abstract = "The following two decision problems capture the complexity of comparing integers or rationals that are succinctly represented in product-of-exponentials notation, or equivalently, via arithmetic circuits using only multiplication and division gates, and integer inputs. Input instance: Four lists of positive integers: a 1,\ldots{}, an \in N+ n; b 1,\ldots{}, bn \in N+ n; c 1,\ldots{}, cm \in N+ m; d 1, \ldots{}, dm \in N+ m; where each of the integers is represented in binary. Problem 1 (equality testing): Decide whether $a_1 b_1 a_2 b_2 \cdots a_n b_n = c_1 d_1 c_2 d_2 \cdots c_m d_m$. Problem 2 (inequality testing): Decide whether $a_1 b_1 a_2 b_2 \cdots a_n b_n \geq c_1 d_1 c_2 d_2 \cdots c_m d_m$. Problem 1 is easily decidable in polynomial time using a simple iterative algorithm. Problem 2 is much harder. We observe that the complexity of Problem 2 is intimately connected to deep conjectures and results in number theory. In particular, if a refined form of the ABC conjecture formulated by Baker in 1998 holds, or if the older Lang-Waldschmidt conjecture (formulated in 1978) on linear forms in logarithms holds, then Problem 2 is decidable in P-time (in the standard Turing model of computation). Moreover, it follows from the best available quantitative bounds on linear forms in logarithms, namely, by Baker and W{\"u}stholz [1993] or Matveev [2000], that if m and n are fixed universal constants then Problem 2 is decidable in P-time (without relying on any conjectures). This latter fact was observed earlier by Shub [1993]. We describe one application: P-time maximum probability parsing for arbitrary stochastic context-free grammars (where \epsilon -rules are allowed).", acknowledgement = ack-nhfb, articleno = "9", fjournal = "ACM Transactions on Computation Theory", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J1190", } @Article{Cook:2014:CCC, author = "Stephen A. Cook and Yuval Filmus and Dai Tri Man L{\^e}", title = "The complexity of the comparator circuit value problem", journal = j-TOCT, volume = "6", number = "4", pages = "15:1--15:??", month = aug, year = "2014", CODEN = "????", DOI = "http://dx.doi.org/10.1145/2635822", ISSN = "1942-3454 (print), 1942-3462 (electronic)", ISSN-L = "1942-3454", bibdate = "Mon Aug 18 17:06:20 MDT 2014", bibsource = "http://www.acm.org/pubs/contents/journals/toct/; http://www.math.utah.edu/pub/tex/bib/toct.bib", abstract = "In 1990, Subramanian [1990] defined the complexity class CC as the set of problems log-space reducible to the comparator circuit value problem (CCV). He and Mayr showed that NL $ \subseteq $ CC $ \subseteq $ P, and proved that in addition to CCV several other problems are complete for CC, including the stable marriage problem, and finding the lexicographically first maximal matching in a bipartite graph. Although the class has not received much attention since then, we are interested in CC because we conjecture that it is incomparable with the parallel class NC which also satisfies NL $ \subseteq $ NC $ \subseteq $ P, and note that this conjecture implies that none of the CC-complete problems has an efficient polylog time parallel algorithm. We provide evidence for our conjecture by giving oracle settings in which relativized CC and relativized NC are incomparable. We give several alternative definitions of CC, including (among others) the class of problems computed by uniform polynomial-size families of comparator circuits supplied with copies of the input and its negation, the class of problems AC0-reducible to Ccv, and the class of problems computed by uniform AC0 circuits with AXccv gates. We also give a machine model for CC, which corresponds to its characterization as log-space uniform polynomial-size families of comparator circuits. These various characterizations show that CC is a robust class. Our techniques also show that the corresponding function class FCC is closed under composition. The main technical tool we employ is universal comparator circuits. Other results include a simpler proof of NL $ \subseteq $ CC, a more careful analysis showing the lexicographically first maximal matching problem and its variants are CC-complete under AC0 many-one reductions, and an explanation of the relation between the Gale--Shapley algorithm and Subramanian's algorithm for stable marriage. This article continues the previous work of Cook et al. [2011], which focused on Cook-Nguyen style uniform proof complexity, answering several open questions raised in that article.", acknowledgement = ack-nhfb, articleno = "15", fjournal = "ACM Transactions on Computation Theory", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J1190", } @Article{Fellows:2014:FCU, author = "Michael R. Fellows and Bart M. P. Jansen", title = "{FPT} is characterized by useful obstruction sets: Connecting algorithms, kernels, and quasi-orders", journal = j-TOCT, volume = "6", number = "4", pages = "16:1--16:??", month = aug, year = "2014", CODEN = "????", DOI = "http://dx.doi.org/10.1145/2635820", ISSN = "1942-3454 (print), 1942-3462 (electronic)", ISSN-L = "1942-3454", bibdate = "Mon Aug 18 17:06:20 MDT 2014", bibsource = "http://www.acm.org/pubs/contents/journals/toct/; http://www.math.utah.edu/pub/tex/bib/toct.bib", abstract = "Many graph problems were first shown to be fixed-parameter tractable using the results of Robertson and Seymour on graph minors. We show that the combination of finite, computable obstruction sets and efficient order tests is not just one way of obtaining strongly uniform FPT algorithms, but that all of FPT may be captured in this way. Our new characterization of FPT has a strong connection to the theory of kernelization, as we prove that problems with polynomial kernels can be characterized by obstruction sets whose elements have polynomial size. Consequently we investigate the interplay between the sizes of problem kernels and the sizes of the elements of such obstruction sets, obtaining several examples of how results in one area yield new insights in the other. We show how exponential-size minor-minimal obstructions for pathwidth $k$ form the crucial ingredient in a novel or-cross-composition for $k$-Pathwidth, complementing the trivial and-composition that is known for this problem. In the other direction, we show that or-cross-compositions into a parameterized problem can be used to rule out the existence of efficiently generated quasi-orders on its instances that characterize the no-instances by polynomial-size obstructions.", acknowledgement = ack-nhfb, articleno = "16", fjournal = "ACM Transactions on Computation Theory", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J1190", } @Article{Gobel:2014:CCH, author = "Andreas G{\"o}bel and Leslie Ann Goldberg and David Richerby", title = "The complexity of counting homomorphisms to cactus graphs modulo 2", journal = j-TOCT, volume = "6", number = "4", pages = "17:1--17:??", month = aug, year = "2014", CODEN = "????", DOI = "http://dx.doi.org/10.1145/2635825", ISSN = "1942-3454 (print), 1942-3462 (electronic)", ISSN-L = "1942-3454", bibdate = "Mon Aug 18 17:06:20 MDT 2014", bibsource = "http://www.acm.org/pubs/contents/journals/toct/; http://www.math.utah.edu/pub/tex/bib/toct.bib", abstract = "A homomorphism from a graph $G$ to a graph $H$ is a function from $ V(G)$ to $ V(H)$ that preserves edges. Many combinatorial structures that arise in mathematics and in computer science can be represented naturally as graph homomorphisms and as weighted sums of graph homomorphisms. In this article, we study the complexity of counting homomorphisms modulo 2. The complexity of modular counting was introduced by Papadimitriou and Zachos and it has been pioneered by Valiant who famously introduced a problem for which counting modulo 7 is easy but counting modulo 2 is intractable. Modular counting provides a rich setting in which to study the structure of homomorphism problems. In this case, the structure of the graph $H$ has a big influence on the complexity of the problem. Thus, our approach is graph-theoretic. We give a complete solution for the class of cactus graphs, which are connected graphs in which every edge belongs to at most one cycle. Cactus graphs arise in many applications such as the modelling of wireless sensor networks and the comparison of genomes. We show that, for some cactus graphs $H$, counting homomorphisms to $H$ modulo 2 can be done in polynomial time. For every other fixed cactus graph $H$, the problem is complete in the complexity class $ \oplus P$, which is a wide complexity class to which every problem in the polynomial hierarchy can be reduced (using randomised reductions). Determining which $H$ lead to tractable problems can be done in polynomial time. Our result builds upon the work of Faben and Jerrum, who gave a dichotomy for the case in which $H$ is a tree.", acknowledgement = ack-nhfb, articleno = "17", fjournal = "ACM Transactions on Computation Theory", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J1190", } @Article{Watson:2014:ALB, author = "Thomas Watson", title = "Advice Lower Bounds for the Dense Model Theorem", journal = j-TOCT, volume = "7", number = "1", pages = "1:1--1:??", month = dec, year = "2014", CODEN = "????", DOI = "http://dx.doi.org/10.1145/2676659", ISSN = "1942-3454 (print), 1942-3462 (electronic)", ISSN-L = "1942-3454", bibdate = "Tue Jan 13 17:53:00 MST 2015", bibsource = "http://www.acm.org/pubs/contents/journals/toct/; http://www.math.utah.edu/pub/tex/bib/toct.bib", abstract = "We prove a lower bound on the amount of nonuniform advice needed by black-box reductions for the Dense Model Theorem of Green, Tao, and Ziegler, and of Reingold, Trevisan, Tulsiani, and Vadhan. The latter theorem roughly says that for every distribution $D$ that is $ \delta $-dense in a distribution that is $ \epsilon '$-indistinguishable from uniform, there exists a ``dense model'' for $D$, that is, a distribution that is $ \delta $ -dense in the uniform distribution and is $ \epsilon $-indistinguishable from $D$. This $ \epsilon $-indistinguishability is with respect to an arbitrary small class of functions $F$. For the natural case where $ \epsilon ' \geq \Omega (\epsilon \delta)$ and $ \epsilon \geq \delta O(1)$, our lower bound implies that $ \Omega (\sqrt (1 / \epsilon) \log (1 / \delta) \cdot \log | F |)$ advice bits are necessary for a certain type of reduction that establishes a stronger form of the Dense Model Theorem (and which encompasses all known proofs of the Dense Model Theorem in the literature). There is only a polynomial gap between our lower bound and the best upper bound for this case (due to Zhang), which is $ O ((1 / \epsilon^2) \log (1 / \delta) \cdot \log | F |)$. Our lower bound can be viewed as an analogue of list size lower bounds for list-decoding of error-correcting codes, but for ``dense model decoding'' instead.", acknowledgement = ack-nhfb, articleno = "1", fjournal = "ACM Transactions on Computation Theory", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J1190", } @Article{Awasthi:2014:LLF, author = "Pranjal Awasthi and Madhav Jha and Marco Molinaro and Sofya Raskhodnikova", title = "Limitations of Local Filters of {Lipschitz} and Monotone Functions", journal = j-TOCT, volume = "7", number = "1", pages = "2:1--2:??", month = dec, year = "2014", CODEN = "????", DOI = "http://dx.doi.org/10.1145/2692372.2692373", ISSN = "1942-3454 (print), 1942-3462 (electronic)", ISSN-L = "1942-3454", bibdate = "Tue Jan 13 17:53:00 MST 2015", bibsource = "http://www.acm.org/pubs/contents/journals/toct/; http://www.math.utah.edu/pub/tex/bib/toct.bib", abstract = "We study local filters for two properties of functions of the form $ f : \{ 0, 1 d \} \to R $: the Lipschitz property and monotonicity. A local filter with additive error a is a randomized algorithm that is given black-box access to a function $f$ and a query point $x$ in the domain of $f$. It outputs a value $f$ (x) such that (i) the reconstructed function $ f(x)$ satisfies the property (in our case, is Lipschitz or monotone) and (ii) if the input function $f$ satisfies the property, then for every point $x$ in the domain (with high constant probability) the reconstructed value $ F(x)$ differs from $ f(x)$ by at most $a$. Local filters were introduced by Saks and Seshadhri [2010]. The relaxed definition we study is due to Bhattacharyya et al. [2012], except that we further relax it by allowing additive error. Local filters for Lipschitz and monotone functions have applications to areas such as data privacy. We show that every local filter for Lipschitz or monotone functions runs in time exponential in the dimension d, even when the filter is allowed significant additive error. Prior lower bounds (for local filters with no additive error, that is, with $ a = 0$) applied only to a more restrictive class of filters, for example, nonadaptive filters. To prove our lower bounds, we construct families of hard functions and show that lookups of a local filter on these functions are captured by a combinatorial object that we call a $c$-connector. Then we present a lower bound on the maximum outdegree of a $c$-connector and show that it implies the desired bounds on the running time of local filters. Our lower bounds, in particular, imply the same bound on the running time for a class of privacy mechanisms.", acknowledgement = ack-nhfb, articleno = "2", fjournal = "ACM Transactions on Computation Theory", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J1190", } @Article{Greco:2014:CNC, author = "Gianluigi Greco and Enrico Malizia and Luigi Palopoli and Francesco Scarcello", title = "The Complexity of the Nucleolus in Compact Games", journal = j-TOCT, volume = "7", number = "1", pages = "3:1--3:??", month = dec, year = "2014", CODEN = "????", DOI = "http://dx.doi.org/10.1145/2692372.2692374", ISSN = "1942-3454 (print), 1942-3462 (electronic)", ISSN-L = "1942-3454", bibdate = "Tue Jan 13 17:53:00 MST 2015", bibsource = "http://www.acm.org/pubs/contents/journals/toct/; http://www.math.utah.edu/pub/tex/bib/toct.bib", abstract = "The nucleolus is a well-known solution concept for coalitional games to fairly distribute the total available worth among the players. The nucleolus is known to be NP -hard to compute over compact coalitional games, that is, over games whose functions specifying the worth associated with each coalition are encoded in terms of polynomially computable functions over combinatorial structures. In particular, hardness results have been exhibited over minimum spanning tree games, threshold games, and flow games. However, due to its intricate definition involving reasoning over exponentially many coalitions, a nontrivial upper bound on its complexity was missing in the literature and looked for. This article faces this question and precisely characterizes the complexity of the nucleolus, by exhibiting an upper bound that holds on any class of compact games, and by showing that this bound is tight even on the (structurally simple) class of graph games. The upper bound is established by proposing a variant of the standard linear-programming based algorithm for nucleolus computation and by studying a framework for reasoning about succinctly specified linear programs, which are contributions of interest in their own. The hardness result is based on an elaborate combinatorial reduction, which is conceptually relevant for it provides a ``measure'' of the computational cost to be paid for guaranteeing voluntary participation to the distribution process. In fact, the pre-nucleolus is known to be efficiently computable over graph games, with this solution concept being defined as the nucleolus but without guaranteeing that each player is granted with it at least the worth she can get alone, that is, without collaborating with the other players. Finally, this article identifies relevant tractable classes of coalitional games, based on the notion of type of a player. Indeed, in most applications where many players are involved, it is often the case that such players do belong in fact to a limited number of classes, which is known in advance and may be exploited for computing the nucleolus in a fast way.", acknowledgement = ack-nhfb, articleno = "3", fjournal = "ACM Transactions on Computation Theory", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J1190", } @Article{Kratsch:2014:KLB, author = "Stefan Kratsch and Marcin Pilipczuk and Ashutosh Rai and Venkatesh Raman", title = "Kernel Lower Bounds using Co-Nondeterminism: Finding Induced Hereditary Subgraphs", journal = j-TOCT, volume = "7", number = "1", pages = "4:1--4:??", month = dec, year = "2014", CODEN = "????", DOI = "http://dx.doi.org/10.1145/2691321", ISSN = "1942-3454 (print), 1942-3462 (electronic)", ISSN-L = "1942-3454", bibdate = "Tue Jan 13 17:53:00 MST 2015", bibsource = "http://www.acm.org/pubs/contents/journals/toct/; http://www.math.utah.edu/pub/tex/bib/toct.bib", abstract = "This work further explores the applications of co-nondeterminism for showing kernelization lower bounds. The only known example prior to this work excludes polynomial kernelizations for the so-called Ramsey problem of finding an independent set or a clique of at least $k$ vertices in a given graph [Kratsch 2012]. We study the more general problem of finding induced subgraphs on $k$ vertices fulfilling some hereditary property $ \Pi $, called $ \Pi $-Induced Subgraph. The problem is NP-hard for all nontrivial choices of $ \Pi $ by a classic result of Lewis and Yannakakis [1980]. The parameterized complexity of this problem was classified by Khot and Raman [2002] depending on the choice of $ \Pi $. The interesting cases for kernelization are for $ \Pi $ containing all independent sets and all cliques, since the problem is trivially polynomial time solvable or W[1]-hard otherwise. Our results are twofold. Regarding $ \Pi $-Induced Subgraph, we show that for a large choice of natural graph properties $ \Pi $, including chordal, perfect, cluster, and cograph, there is no polynomial kernel with respect to $k$. This is established by two theorems, each one capturing different (but not necessarily exclusive) sets of properties: one using a co-nondeterministic variant of OR-cross-composition and one by a polynomial parameter transformation from Ramsey. Additionally, we show how to use improvement versions of NP-hard problems as source problems for lower bounds, without requiring their NP-hardness. For example, for $ \Pi $-Induced Subgraph our compositions may assume existing solutions of size $ k - 1$. This follows from the more general fact that source problems for OR-(cross-)compositions need only be NP-hard under co-nondeterministic reductions. We believe this to be useful for further lower-bound proofs, for example, since improvement versions simplify the construction of a disjunction (OR) of instances required in compositions. This adds a second way of using co-nondeterminism for lower bounds.", acknowledgement = ack-nhfb, articleno = "4", fjournal = "ACM Transactions on Computation Theory", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J1190", } @Article{Filmus:2015:ELB, author = "Yuval Filmus and Toniann Pitassi and Rahul Santhanam", title = "Exponential Lower Bounds for {AC$0$-Frege} Imply Superpolynomial {Frege} Lower Bounds", journal = j-TOCT, volume = "7", number = "2", pages = "5:1--5:??", month = may, year = "2015", CODEN = "????", DOI = "http://dx.doi.org/10.1145/2656209", ISSN = "1942-3454 (print), 1942-3462 (electronic)", ISSN-L = "1942-3454", bibdate = "Tue May 12 06:02:22 MDT 2015", bibsource = "http://www.acm.org/pubs/contents/journals/toct/; http://www.math.utah.edu/pub/tex/bib/toct.bib", abstract = "We give a general transformation that turns polynomial-size Frege proofs into subexponential-size AC$^0$-Frege proofs. This indicates that proving truly exponential lower bounds for AC$^0$-Frege is hard, as it is a long-standing open problem to prove superpolynomial lower bounds for Frege. Our construction is optimal for proofs of formulas of unbounded depth. As a consequence of our main result, we are able to shed some light on the question of automatizability for bounded-depth Frege systems. First, we present a simpler proof of the results of Bonet et al. showing that under cryptographic assumptions, bounded-depth Frege proofs are not automatizable. Second, we show that because our proof is more general, under the right cryptographic assumptions, it could resolve the automatizability question for lower-depth Frege systems.", acknowledgement = ack-nhfb, articleno = "5", fjournal = "ACM Transactions on Computation Theory", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J1190", } @Article{Blaser:2015:SCT, author = "Markus Bl{\"a}ser and Bodo Manthey", title = "Smoothed Complexity Theory", journal = j-TOCT, volume = "7", number = "2", pages = "6:1--6:??", month = may, year = "2015", CODEN = "????", DOI = "http://dx.doi.org/10.1145/2656210", ISSN = "1942-3454 (print), 1942-3462 (electronic)", ISSN-L = "1942-3454", bibdate = "Tue May 12 06:02:22 MDT 2015", bibsource = "http://www.acm.org/pubs/contents/journals/toct/; http://www.math.utah.edu/pub/tex/bib/toct.bib", abstract = "Smoothed analysis is a new way of analyzing algorithms introduced by Spielman and Teng. Classical methods like worst-case or average-case analysis have accompanying complexity classes, such as P and Avg-P, respectively. Whereas worst-case or average-case analysis give us a means to talk about the running time of a particular algorithm, complexity classes allow us to talk about the inherent difficulty of problems. Smoothed analysis is a hybrid of worst-case and average-case analysis and compensates some of their drawbacks. Despite its success for the analysis of single algorithms and problems, there is no embedding of smoothed analysis into computational complexity theory, which is necessary to classify problems according to their intrinsic difficulty. We propose a framework for smoothed complexity theory, define the relevant classes, and prove some first hardness results (of bounded halting and tiling) and tractability results (binary optimization problems, graph coloring, satisfiability) within this framework.", acknowledgement = ack-nhfb, articleno = "6", fjournal = "ACM Transactions on Computation Theory", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J1190", } @Article{Chen:2015:FCC, author = "Hubie Chen and Moritz M{\"u}ller", title = "The Fine Classification of Conjunctive Queries and Parameterized Logarithmic Space", journal = j-TOCT, volume = "7", number = "2", pages = "7:1--7:??", month = may, year = "2015", CODEN = "????", DOI = "http://dx.doi.org/10.1145/2751316", ISSN = "1942-3454 (print), 1942-3462 (electronic)", ISSN-L = "1942-3454", bibdate = "Tue May 12 06:02:22 MDT 2015", bibsource = "http://www.acm.org/pubs/contents/journals/toct/; http://www.math.utah.edu/pub/tex/bib/toct.bib", abstract = "We perform a fundamental investigation of the complexity of conjunctive query evaluation from the perspective of parameterized complexity. We classify sets of Boolean conjunctive queries according to the complexity of this problem. Previous work showed that a set of conjunctive queries is fixed-parameter tractable precisely when the set is equivalent to a set of queries having bounded treewidth. We present a fine classification of query sets up to parameterized logarithmic space reduction. We show that, in the bounded treewidth regime, there are three complexity degrees and that the properties that determine the degree of a query set are bounded pathwidth and bounded tree depth. We also engage in a study of the two higher degrees via logarithmic space machine characterizations and complete problems. Our work yields a significantly richer perspective on the complexity of conjunctive queries and, at the same time, suggests new avenues of research in parameterized complexity.", acknowledgement = ack-nhfb, articleno = "7", fjournal = "ACM Transactions on Computation Theory", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J1190", } @Article{Komarath:2015:PEB, author = "Balagopal Komarath and Jayalal Sarma", title = "Pebbling, Entropy, and Branching Program Size Lower Bounds", journal = j-TOCT, volume = "7", number = "2", pages = "8:1--8:??", month = may, year = "2015", CODEN = "????", DOI = "http://dx.doi.org/10.1145/2751320", ISSN = "1942-3454 (print), 1942-3462 (electronic)", ISSN-L = "1942-3454", bibdate = "Tue May 12 06:02:22 MDT 2015", bibsource = "http://www.acm.org/pubs/contents/journals/toct/; http://www.math.utah.edu/pub/tex/bib/toct.bib", abstract = "We contribute to the program of proving lower bounds on the size of branching programs solving the Tree Evaluation Problem introduced by Cook et al. [2012]. Proving a superpolynomial lower bound for the size of nondeterministic thrifty branching programs would be an important step toward separating NL from P using the tree evaluation problem. First, we show that Read-Once Nondeterministic Thrifty BPs are equivalent to whole black-white pebbling algorithms, thus showing a tight lower bound (ignoring polynomial factors) for this model. We then introduce a weaker restriction of nondeterministic thrifty branching programs called Bitwise Independence. The best known [Cook et al. 2012] nondeterministic thrifty branching programs (of size $ O(k^{h / 2 + 1})$) for the tree evaluation problem are Bitwise Independent. As our main result, we show that any Bitwise Independent Nondeterministic Thrifty Branching Program solving $ {\rm BT}_2 (h, k)$ must have at least $ (k2)^{h / 2}$ states. Prior to this work, lower bounds were known for nondeterministic thrifty branching programs only for fixed heights $ h = 2, 3, 4$ [Cook et al. 2012]. We prove our results by associating a fractional black-white pebbling strategy with any bitwise independent nondeterministic thrifty branching program solving the Tree Evaluation Problem. Such a connection was not known previously, even for fixed heights. Our main technique is the entropy method introduced by Jukna and Z{\'a}k [2001] originally in the context of proving lower bounds for read-once branching programs. We also show that the previous lower bounds known [Cook et al. 2012] for deterministic branching programs for the Tree Evaluation Problem can be obtained using this approach. Using this method, we also show tight lower bounds for any $k$-way deterministic branching program solving the Tree Evaluation Problem when the instances are restricted to have the same group operation in all internal nodes.", acknowledgement = ack-nhfb, articleno = "8", fjournal = "ACM Transactions on Computation Theory", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J1190", } @Article{ODonnell:2015:HMM, author = "Ryan O'Donnell and Yi Wu and Yuan Zhou", title = "Hardness of {Max-2Lin} and {Max-3Lin} over Integers, Reals, and Large Cyclic Groups", journal = j-TOCT, volume = "7", number = "2", pages = "9:1--9:??", month = may, year = "2015", CODEN = "????", DOI = "http://dx.doi.org/10.1145/2751322", ISSN = "1942-3454 (print), 1942-3462 (electronic)", ISSN-L = "1942-3454", bibdate = "Tue May 12 06:02:22 MDT 2015", bibsource = "http://www.acm.org/pubs/contents/journals/toct/; http://www.math.utah.edu/pub/tex/bib/toct.bib", abstract = "In 1997, H{\aa}stad showed NP-hardness of $ (1 - \epsilon, 1 / q + \delta)$-approximating Max-3Lin($ Z_q$); however, it was not until 2007 that Guruswami and Raghavendra were able to show NP-hardness of $ (1 - \epsilon, \delta)$-approximating Max-3Lin($Z$). In 2004, Khot--Kindler--Mossel--O'Donnell showed UG-hardness of $ (1 - \epsilon, \delta)$-approximating Max-2Lin($ Z_q$) for $ q = q (\epsilon, \delta)$ a sufficiently large constant; however, achieving the same hardness for Max-2Lin($Z$) was given as an open problem in Raghavendra's 2009 thesis. In this work, we show that fairly simple modifications to the proofs of the Max-3Lin($ Z_q$) and Max-2Lin($ Z_q$) results yield optimal hardness results over $Z$. In fact, we show a kind of ``bicriteria'' hardness: Even when there is a $ (1 - \epsilon)$-good solution over $Z$, it is hard for an algorithm to find a $ \delta $-good solution over $Z$, $R$, or $ Z_m$ for any $ m \geq q (\epsilon, \delta)$ of the algorithm's choosing.", acknowledgement = ack-nhfb, articleno = "9", fjournal = "ACM Transactions on Computation Theory", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J1190", }