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

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^*(eps^{-0.617}) where eps is the fraction of assignments that are satisfying. This improves significantly over the trivial sampling bound of expected Theta^*(eps^{-1}), and on all previous algorithms whenever eps = Omega(0.708^n). We also consider algorithms for 3-SAT with an eps fraction of satisfying assignments, and prove that it can be solved in O^*(eps^{-2.27}) deterministic time, and in O^*(eps^{-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.