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We present a Monte Carlo algorithm for Hamiltonicity detection in an $n$-vertex undirected graph running in $O^*(2^{frac{3}{4}n})$ 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 $O^*(2^{frac{1}{2}n})$ 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 result. Our main technical contribution is a new reduction inspired by the algebraic sieving method for $k$-Path (Koutis ICALP 2008, Williams IPL 2009). We transform the Hamiltonicity instance into many smaller Cycle Cover instances in which we are left to count weighted cycle covers over a finite field of characteristic two. We next adopt and apply the determinant summation technique for Exact Set Covers (Bj"orklund STACS 2010).

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