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Determinant sums for undirected Hamiltonicity

Publiceringsår: 2010
Språk: Engelska
Sidor: 173-182
Publikation/Tidskrift/Serie: 2010 IEEE 51st Annual Symposium On Foundations Of Computer Science
Dokumenttyp: Konferensbidrag
Förlag: IEEE--Institute of Electrical and Electronics Engineers Inc.


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.


  • Computer Science


51st Annual IEEE Symposium on Foundations of Computer Science (FOCS 2010)
  • Exact algorithms
  • Algorithms-lup-obsolete
  • ISSN: 0272-5428
  • ISBN: 978-0-7695-4244-7

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