Abstract
We show that for k ≥ 5, the PPSZ algorithm for k-SAT runs exponentially faster if there is an exponential number of satisfying assignments. More precisely, we show that for every k≥ 5, there is a strictly increasing function f: [0,1] → R with f(0) = 0 that has the following property. If F is a k-CNF formula over n variables and |sat(F)| = 2δ n solutions, then PPSZ finds a satisfying assignment with probability at least 2−ckn −o(n) + f(δ) n. Here, 2−ckn −o(n) is the success probability proved by Paturi et al. [11] for k-CNF formulas with a unique satisfying assignment.
Our proof rests on a combinatorial lemma: given a set S ⊆ { 0,1} n, we can partition { 0,1} n into subcubes such that each subcube B contains exactly one element of S. Such a partition B induces a distribution on itself, via Pr [B] = |B| / 2n for each B ∈ B. We are interested in partitions that induce a distribution of high entropy. We show that, in a certain sense, the worst case (minS: |S| = s maxB H(B)) is achieved if S is a Hamming ball. This lemma implies that every set S of exponential size allows a partition of linear entropy. This in turn leads to an exponential improvement of the success probability of PPSZ.
- Sven Baumer and Rainer Schuler. 2003. Improving a probabilistic 3-SAT algorithm by dynamic search and independent clause pairs. In Proceedings of the 6th International Conference on Theory and Applications of Satisfiability Testing (SAT’03) (Lecture Notes in Computer Science), Enrico Giunchiglia and Armando Tacchella (Eds.), vol. 2919. Springer, 150--161.Google Scholar
- Béla Bollobás. 1986. Combinatorics: Set Systems, Hypergraphs, Families of Vectors, and Combinatorial Probability. Cambridge University Press, New York, NY. Google Scholar
Digital Library
- Chris Calabro, Russell Impagliazzo, Valentine Kabanets, and Ramamohan Paturi. 2008. The complexity of unique k-SAT: An isolation lemma for k-CNFs. J. Comput. Syst. Sci. 74, 3 (2008), 386--393. Google Scholar
Digital Library
- Timon Hertli. 2011. 3-SAT faster and simpler—Unique-SAT bounds for PPSZ hold in general. In Proceedings of the IEEE 52nd Annual Symposium on Foundations of Computer Science (FOCS’11). IEEE, Los Alamitos, CA, 277--284. Google Scholar
Digital Library
- Timon Hertli. 2014. Breaking the PPSZ barrier for unique 3-SAT. In Proceedings of the 41st International Colloquium on Automata, Languages, and Programming (ICALP’14) (Lecture Notes in Computer Science), Javier Esparza, Pierre Fraigniaud, Thore Husfeldt, and Elias Koutsoupias (Eds.), vol. 8572. Springer, 600--611.Google Scholar
Cross Ref
- Timon Hertli, Robin A. Moser, and Dominik Scheder. 2011. Improving PPSZ for 3-SAT using critical variables. In Proceedings of the 19th Annual Symposium on Theoretical Aspects of Computer Science (STACS’11). 237--248.Google Scholar
- Thomas Hofmeister, Uwe Schöning, Rainer Schuler, and Osamu Watanabe. 2002. A probabilistic 3-SAT algorithm further improved. In Proceedings of the 19th Annual Symposium on Theoretical Aspects of Computer Science (STACS’02) (Lecture Notes in Computer Science), Helmut Alt and Afonso Ferreira (Eds.), vol. 2285. Springer, 192--202. Google Scholar
Digital Library
- Kazuo Iwama, Kazuhisa Seto, Tadashi Takai, and Suguru Tamaki. 2010. Improved randomized algorithms for 3-SAT. In Proceedings of the 21st International Symposium on Algorithms and Computation (ISAAC’10) (Lecture Notes in Computer Science), Otfried Cheong, Kyung-Yong Chwa, and Kunsoo Park (Eds.), vol. 6506. Springer, 73--84.Google Scholar
Cross Ref
- Kazuo Iwama and Suguru Tamaki. 2004. Improved upper bounds for 3-SAT. In Proceedings of the 15th Annual ACM-SIAM Symposium on Discrete Algorithms. ACM, New York, 328--329. Google Scholar
Digital Library
- Ramamohan Paturi. {n.d.}. Personal communication.Google Scholar
- Ramamohan Paturi, Pavel Pudlák, Michael E. Saks, and Francis Zane. 2005. An improved exponential-time algorithm for k-SAT. J. ACM 52, 3 (2005), 337--364 (electronic). Google Scholar
Digital Library
- Ramamohan Paturi, Pavel Pudlák, and Francis Zane. 1999. Satisfiability coding lemma. Chicago J. Theoret. Comput. Sci. (1999).Google Scholar
- Daniel Rolf. 2006. Improved bound for the PPSZ/Schöning-Algorithm for 3-SAT. J. Satisfiabil., Boolean Model. Comput. 1 (2006), 111--122.Google Scholar
Cross Ref
- N. Sauer. 1972. On the density of families of sets. J. Combinat. Theory, Ser. A 13, 1 (1972), 145--147.Google Scholar
Cross Ref
- Dominik Scheder and John P. Steinberger. 2017. PPSZ for general k-SAT—Making Hertli’s analysis simpler and 3-SAT faster. In Proceedings of the 32nd Computational Complexity Conference (CCC’17) (LIPIcs), Ryan O’Donnell (Ed.), vol. 79. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 9:1--9:15. Google Scholar
Digital Library
- Uwe Schöning. 1999. A probabilistic algorithm for k-SAT and constraint satisfaction problems. In Proceedings of the 40th Annual Symposium on Foundations of Computer Science. IEEE, Los Alamitos, CA, 410--414. Google Scholar
Digital Library
- Saharon Shelah. 1972. A combinatorial problem: Stability and order for models and theories in infinitary languages. Pacific J. Math. 41, 1 (1972), 247--261.Google Scholar
Cross Ref
Index Terms
PPSZ for k ≥ 5: More Is Better
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