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Algorithmica

, Volume 52, Issue 2, pp 226–249 | Cite as

Exact Algorithms for Exact Satisfiability and Number of Perfect Matchings

  • Andreas Björklund
  • Thore HusfeldtEmail author
Article

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 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 2 n n O(1) and polynomial space. We also show how to count the number of perfect matchings in time O(1.732 n ) 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.

Keywords

Exact algorithms Set cover Set partition Exact satisfability Number of perfect matchings 

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Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Department of Computer ScienceLund UniversityLundSweden

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