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Determinant sums for undirected Hamiltonicity

Author

  • Andreas Björklund

Summary, in English

Abstract in Undetermined
We present a Monte Carlo algorithm for Hamiltonicity detection in an n-vertex undirected graph running in O*(1.657(n)) time. To the best of our knowledge, this is the first superpolynomial improvement on the worst case runtime for the problem since the O*(2(n)) bound established for TSP almost fifty years ago (Bellman 1962, Held and Karp 1962). It answers in part the first open problem in Woeginger's 2003 survey on exact algorithms for NP-hard problems.

For bipartite graphs, we improve the bound to O*(1.414(n)) time. Both the bipartite and the general algorithm can be implemented to use space polynomial in n.

We combine several recently resurrected ideas to get the results. Our main technical contribution is a new reduction inspired by the algebraic sieving method for k-Path (Koutis ICALP 2008, Williams IPL 2009). We introduce the Labeled Cycle Cover Sum in which we are set to count weighted arc labeled cycle covers over a finite field of characteristic two. We reduce Hamiltonicity to Labeled Cycle Cover Sum and apply the determinant summation technique for Exact Set Covers (Bjorklund STACS 2010) to evaluate it.

Publishing year

2010

Language

English

Pages

173-182

Publication/Series

2010 IEEE 51st Annual Symposium On Foundations Of Computer Science

Document type

Conference paper

Publisher

IEEE - Institute of Electrical and Electronics Engineers Inc.

Topic

  • Computer Science

Conference name

51st Annual IEEE Symposium on Foundations of Computer Science (FOCS 2010)

Conference date

2010-10-23 - 2010-10-26

Conference place

Las Vegas, United States

Status

Published

Project

  • Exact algorithms

Research group

  • Algorithms

ISBN/ISSN/Other

  • ISSN: 0272-5428
  • ISBN: 978-0-7695-4244-7