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 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.
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Björklund, A., Husfeldt, T. (2006). Exact Algorithms for Exact Satisfiability and Number of Perfect Matchings. In: Bugliesi, M., Preneel, B., Sassone, V., Wegener, I. (eds) Automata, Languages and Programming. ICALP 2006. Lecture Notes in Computer Science, vol 4051. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11786986_48
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DOI: https://doi.org/10.1007/11786986_48
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