Solving and Sampling with Many Solutions: Satisfiability and Other Hard Problems

Jean Cardinal, Jerri Nummenpalo, Emo Welzl

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 $\varepsilon = \Omega(0.708^n)$. 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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