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Determinant Sums fo...
Determinant Sums for Undirected Hamiltonicity
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- Björklund, Andreas (författare)
- Lund University,Lunds universitet,Institutionen för datavetenskap,Institutioner vid LTH,Lunds Tekniska Högskola,Department of Computer Science,Departments at LTH,Faculty of Engineering, LTH
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(creator_code:org_t)
- Society for Industrial & Applied Mathematics (SIAM), 2014
- 2014
- Engelska.
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Ingår i: SIAM Journal on Computing. - : Society for Industrial & Applied Mathematics (SIAM). - 0097-5397 .- 1095-7111. ; 43:1, s. 280-299
- Relaterad länk:
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Abstract
Ämnesord
Stäng
- We 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 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 $O^*(\sqrt{2}^n)\subset O(1.415^{n})$ 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 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.
Ämnesord
- NATURVETENSKAP -- Data- och informationsvetenskap -- Datavetenskap (hsv//swe)
- NATURAL SCIENCES -- Computer and Information Sciences -- Computer Sciences (hsv//eng)
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