We present a Monte Carlo algorithm for Hamiltonicity detection in an
n-vertex undirected graph running in O∗(1.657n) time. To the best of
our knowledge, this is the first superpolynomial improvement on the worst case
runtime for the problem since the O∗(2n) 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∗(1.414n) 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 (Bj\"orklund STACS 2010) to evaluate it.Comment: To appear at IEEE FOCS 201