Abstract
We investigate the complexity of satisfiability problems parameterized by the size of the solution set compared to all solution candidates. Our main result is a uniform sampling algorithm for satisfying assignments of 2-CNF formulas that runs in expected time \(O^*(\varepsilon ^{-0.617})\) where \(\varepsilon \) is the fraction of assignments that are satisfying. This improves significantly over the trivial sampling bound of expected \(\Theta ^*(\varepsilon ^{-1})\), and on all previous algorithms whenever \(\varepsilon = \Omega (0.708^n)\), where n is the number of variables. We also consider algorithms for 3-SAT with an \(\varepsilon \) fraction of satisfying assignments, and prove that we can output a satisfying assignment in \(O^*(\varepsilon ^{-0.936})\) randomized time, and sample uniformly a satisfying assignment in time \(O^*(\varepsilon ^{-0.908}1.021^n)\). In the end we also present sampling results in the cases of 1-IN-3-SAT, monotone bounded degree 3-SAT and planar k-SAT.
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Notes
This is equivalent to finding a 1-maximal matching in a graph: first find a maximal matching and then find a maximal set of independent augmenting paths of length 3 and augment them.
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Acknowledgements
We would like to thank Noga Alon and József Solymosi for discussions on the problem. We also thank the reviewers of IPEC 2017 and Algorithmica for valuable remarks that improved the exposition.
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This work started at the 2016 Gremo Workshop on Open Problems (GWOP), on June 6–10 at St. Niklausen, OW, Switzerland. A preliminary version was presented at the the 12th International Symposium on Parameterized and Exact Computation (IPEC 2017). The current version contains additional results on 3-SAT problems, but does not include the results on deterministic algorithms for 3-SAT and vertex cover that were presented at the conference.
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Cardinal, J., Nummenpalo, J. & Welzl, E. Solving and Sampling with Many Solutions. Algorithmica 82, 1474–1489 (2020). https://doi.org/10.1007/s00453-019-00654-w
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DOI: https://doi.org/10.1007/s00453-019-00654-w