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Solving and Sampling with Many Solutions: Satisfiability and Other Hard Problems (1708.01122v1)

Published 3 Aug 2017 in cs.DM

Abstract: We investigate parameterizing hard combinatorial problems 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 $\varepsilon = \Omega(0.708n)$. We also consider algorithms for 3-SAT with an $\varepsilon$ fraction of satisfying assignments, and prove that it can be solved in $O*(\varepsilon{-2.27})$ deterministic time, and in $O*(\varepsilon{-0.936})$ randomized time. Finally, to further demonstrate the applicability of this framework, we also explore how similar techniques can be used for vertex cover problems.

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