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Below All Subsets for Some Permutational Counting Problems

Author Andreas Björklund

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Andreas Björklund

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Andreas Björklund. Below All Subsets for Some Permutational Counting Problems. In 15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 53, pp. 17:1-17:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016) https://doi.org/10.4230/LIPIcs.SWAT.2016.17

Abstract

We show that the two problems of computing the permanent of an n*n matrix of poly(n)-bit integers and counting the number of Hamiltonian cycles in a directed n-vertex multigraph with exp(poly(n)) edges can be reduced to relatively few smaller instances of themselves. In effect we derive the first deterministic algorithms for these two problems that run in o(2^n) time in the worst case. Classic poly(n)2^n time algorithms for the two problems have been known since the early 1960's.
Our algorithms run in 2^{n-Omega(sqrt{n/log(n)})} time.

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Keywords
  • Matrix Permanent
  • Hamiltonian Cycles
  • Asymmetric TSP

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