Determinant sums for undirected Hamiltonicity

Björklund, Andreas (author): Lund University,Lunds universitet,Institutionen för datavetenskap,Institutioner vid LTH,Lunds Tekniska Högskola,Department of Computer Science,Departments at LTH,Faculty of Engineering, LTH

(creator_code:org_t)

2010
2010
English.
In: 2010 IEEE 51st Annual Symposium On Foundations Of Computer Science. - 0272-5428. - 9780769542447 ; , s. 173-182

Related links:: http://dx.doi.org/10...; show more...; https://lup.lub.lu.s...; https://doi.org/10.1...; show less...

Abstract Subject headings

Abstract in UndeterminedWe 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.

About the subject

NATURAL SCIENCES: NATURAL SCIENCES; and Computer and Inf ...; and Computer Science ...

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