Improved Bounds for Exact Counting of Satisfiability Solutions

An algorithm is presented for exactly solving (in fact, counting) the number of maximum weight satisfying assignments of a 2-SAT formula. The worst case running time of O(1.2461) for formulas with n variables improves on the previous bound of O(1.2561) by Dahllöf, Jonsson, and Wahlström. The weighted 2-SAT counting algorithm can be applied to obtain faster algorithms for combinatorial counting problems, including those of counting maximum weighted independent sets and weighted set packings. The above result when combined with a better partitioning technique for domains, leads to improved running times for counting the number of solutions of binary constraint satisfaction problems for all domain sizes. For large domain size d we approach O((0.6009d)) improving their previous best bound of O((0.6224d)). We further improve this bound for counting 3-colorings in a graph. We also present an algorithm for exactly counting weighted MAX 2-SAT assignments. Improving the trivial bound (even for the decision version) for MAX 2-SAT has been explicitly stated as a open problem ([28, 1, 22]). For a 2-SAT formula F , we have a worst case running time of O∗(2n(1−1/(d̃(F )−1))), where d̃(F ) is the average degree in the constraint graph defined by F . Our algorithm, together with its analysis is much simpler and avoids some tedious enumerations present in previous results. We use α−gadget introduced by Trevisan, Sorkin, Sudan, and Williamson to get the same upper bound for problems like MAX 3-SAT, MAX CUT. We also introduce a notion of strict (α, β)−gadget to provide a framework that allows composition of gadgets. This framework allows us to obtain the same upper bound for MAX NAE-k-SAT, MAX k-SAT, MAX k-LIN-2. In fact, any problem having a strict (α, β)−gadget reduction to MAX 2-SAT has the same upper bound. This opens the possibility of obtaining better bounds for many other problems. Some of the results appeared in a longer version as ECCC Report TR05-033 [16] Research supported in part by NSF Grant CCR-0209099