Exact Algorithms for Exact Satisfiability and Number of Perfect Matchings

Abstract

AbstractWe present exact algorithms with exponential running times for variants of n-element set cover problems, based on divide-and-conquer and on inclusion–exclusion characterisations.We show that the Exact Satisfiability problem of size l with m clauses can be solved in time 2m l O(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 2n n O(1) and polynomial space. We also show how to count the number of perfect matchings in time O(1.732n) and exponential space.Using the same techniques we show how to compute Chromatic Number of an n-vertex graph in time O(2.4423n) and polynomial space, or time O(2.3236n) and exponential space.KeywordsPerfect MatchingsChromatic NumberExact AlgorithmHamiltonian CycleColour ClassThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.