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

TL;DR

This paper develops efficient algorithms for sampling and solving hard combinatorial problems like SAT and vertex cover, especially when solutions are sparse, improving over previous methods.

Contribution

It introduces a novel uniform sampling algorithm for 2-CNF formulas with improved expected runtime based on the solution fraction, and extends techniques to 3-SAT and vertex cover problems.

Findings

Sampling 2-CNF solutions in expected $O^*( ext{epsilon}^{-0.617})$ time

Deterministic 3-SAT solving in $O^*( ext{epsilon}^{-2.27})$ time

Randomized 3-SAT solving in $O^*( ext{epsilon}^{-0.936})$ time

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