2010/08/03 by Andreas Björklund, Björklund, Andreas · 9 citations
Computer Science · Mathematics · #Advanced Graph Theory Research #Complexity and Algorithms in Graphs #Data Structures and Algorithms (cs.DS) #FOS: Computer and information sciences #Markov Chains and Monte Carlo Methods #cs.DS
paper · pdf · doi:10.48550/arxiv.1008.0541
To appear at IEEE FOCS 2010
arxiv created 2010/08/03 · openalex publication_date 2010/08/03 · arxiv updated 2010/08/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 (Björklund STACS 2010) to evaluate it.