Determinant Sums for Undirected Hamiltonicity

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 the traveling salesman problem (TSP) over 50 years ago [R. Bellman, J. Assoc. Comput. Mach., 9 (1962), pp. 61--63], [M. Held and R. M. Karp, J. Soc. Indust. Appl. Math., 10 (1962), pp. 196--210]. (O^*(f(n)) suppresses polylogarithmic functions in $f(n)$). 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^*(√2ⁿ)⊂ O(1.415ⁿ) 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 algebraic characterization of Hamiltonian graphs. We introduce an extension of Hamiltonicity called Labeled Hamiltonicity and relate it to a Labeled Cycle Cover Sum in which we are set to count weighted arc labeled cycle covers over a finite field of characteristic two. The Labeled Cycle Cover Sum can be evaluated efficiently via determinants.