Algorithms for Counting 2-SatSolutions and Colorings with Applications

An algorithm is presented for exactly solving (in fact, counting) the number of maximum weight satisfying assignments of a 2-Cnf formula. The worst case running time of O(1.246) for formulas with n variables improves on the previous bound of O(1.256) by Dahllöf, Jonsson, and Wahlström. The algorithm uses only polynomial space. As a consequence we get an O(1.246) time algorithm for counting maximum weighted independent sets. 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 approachO((0.601d)) improving the previous best bound of O((0.622d)). We further improve this bound for counting 3-colorings in a graph. The upper bound of O(1.770) for graphs with n vertices improves on the previous bound of O(1.788) by Angelsmark and Jonsson.