License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ICALP.2017.85
URN: urn:nbn:de:0030-drops-74144
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2017/7414/
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Pudlák, Pavel ; Scheder, Dominik ; Talebanfard, Navid

Tighter Hard Instances for PPSZ

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LIPIcs-ICALP-2017-85.pdf (0.6 MB)

Abstract

We construct uniquely satisfiable k-CNF formulas that are hard for the PPSZ algorithm, the currently best known algorithm solving k-SAT. This algorithm tries to generate a satisfying assignment by picking a random variable at a time and attempting to derive its value using some inference heuristic and otherwise assigning a random value. The "weak PPSZ" checks all subformulas of a given size to derive a value and the "strong PPSZ" runs resolution with width bounded by some given function. Firstly, we construct graph-instances on which "weak PPSZ" has savings of at most (2 + epsilon)/k; the saving of an algorithm on an input formula with n variables is the largest gamma such that the algorithm succeeds (i.e. finds a satisfying assignment) with probability at least 2^{- (1 - gamma) n}. Since PPSZ (both weak and strong) is known to have savings of at least (pi^2 + o(1))/6k, this is optimal up to the constant factor. In particular, for k=3, our upper bound is 2^{0.333... n}, which is fairly close to the lower bound 2^{0.386... n} of Hertli [SIAM J. Comput.'14]. We also construct instances based on linear systems over F_2 for which strong PPSZ has savings of at most O(log(k)/k). This is only a log(k) factor away from the optimal bound. Our constructions improve previous savings upper bound of O((log^2(k))/k) due to Chen et al. [SODA'13].

BibTeX - Entry

@InProceedings{pudlk_et_al:LIPIcs:2017:7414,
  author =	{Pavel Pudl{\'a}k and Dominik Scheder and Navid Talebanfard},
  title =	{{Tighter Hard Instances for PPSZ}},
  booktitle =	{44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)},
  pages =	{85:1--85:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-041-5},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{80},
  editor =	{Ioannis Chatzigiannakis and Piotr Indyk and Fabian Kuhn and Anca Muscholl},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2017/7414},
  URN =		{urn:nbn:de:0030-drops-74144},
  doi =		{10.4230/LIPIcs.ICALP.2017.85},
  annote =	{Keywords: k-SAT, Strong Exponential Time Hypothesis, PPSZ, Resolution}
}

Keywords: k-SAT, Strong Exponential Time Hypothesis, PPSZ, Resolution
Collection: 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)
Issue Date: 2017
Date of publication: 07.07.2017

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