Exact Algorithms for Exact Satisfiability and Number of Perfect Matchings

Abstract

We present exact algorithms with exponential running times for variants of n-element set cover problems, based on divide-and-conquer and on inclusion–exclusion characterizations. We show that the Exact Satisfiability problem of size l with m clauses can be solved in time 2 m lO(1) and polynomial space. The same bounds hold for counting the number of solutions. As a special case, we can count the number of perfect matchings in an n-vertex graph in time 2nn O(1) and polynomial space. We also show how to count the number of perfect matchings in time O(1.732n) and exponential space. We give a number of examples where the running time can be further improved if the hypergraph corresponding to the set cover instance has low pathwidth. This yields exponential-time algorithms for counting k-dimensional matchings, Exact Uniform Set Cover, Clique Partition, and Minimum Dominating Set in graphs of degree at most three. We extend the analysis to a number of related problems such as TSP and Chromatic Number.