Pudlák, P., Scheder, D. and Talebanfard, N. (2017) Tighter hard instances for PPSZ. In: Chatzigiannakis, I., Indyk, P., Kuhn, F. and Muscholl, A., (eds.) 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017), 10-14 Jul 2017, Warsaw, Poland. Leibniz International Proceedings in Informatics (LIPIcs), 80 . Schloss Dagstuhl - Leibniz-Zentrum für Informatik , 85:1-85:13. ISBN 9783959770415
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].
Metadata
Item Type: | Proceedings Paper |
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Authors/Creators: |
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Editors: |
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Copyright, Publisher and Additional Information: | © 2017 Pavel Pudlák, Dominik Scheder, and Navid Talebanfard. Licensed under Creative Commons License CC-BY (http://creativecommons.org/licenses/by/3.0/) |
Keywords: | k-SAT; Strong Exponential Time Hypothesis; PPSZ; Resolution |
Dates: |
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Institution: | The University of Sheffield |
Academic Units: | The University of Sheffield > Faculty of Engineering (Sheffield) > Department of Computer Science (Sheffield) |
Depositing User: | Symplectic Sheffield |
Date Deposited: | 18 Oct 2023 15:44 |
Last Modified: | 18 Oct 2023 15:44 |
Status: | Published |
Publisher: | Schloss Dagstuhl - Leibniz-Zentrum für Informatik |
Series Name: | Leibniz International Proceedings in Informatics (LIPIcs) |
Refereed: | Yes |
Identification Number: | 10.4230/LIPIcs.ICALP.2017.85 |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:204343 |