The Traveling Salesman Problem for Cubic Graphs

Eppstein, David

doi:10.7155/jgaa.00137

Cited by 84 publications

(95 citation statements)

References 10 publications

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“…Broersma et al [8] proved that Hamiltonicity in claw-free graphs has an O(1.682 n ) time algorithm. Iwama and Nakashima [16], improving slightly on Eppstein [12], showed that the TSP in cubic graphs admits an O(1.251 n ) time algorithm. In graphs of maximum degree four, Gebauer [13] described how to count the Hamiltonian cycles in O(1.733 n ) time.…”

Section: Previous Workmentioning

confidence: 98%

Determinant Sums for Undirected Hamiltonicity

Björklund¹

2014

SIAM J. Comput.

118

172

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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. improve the bound to O * ( √ 2 n ) ⊂ O(1.415 n ) time. Both the bipartite and the general algorithmcan 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.

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Section: Previous Workmentioning

confidence: 98%

Determinant Sums for Undirected Hamiltonicity

Björklund¹

2014

SIAM J. Comput.

118

172

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“…From recurrences (2)(3)(4)(5)(6)(7)(8), P (k) ≤ c k ≤ c (1+α 3 )n , where c = c(β, α 2 , α 3 ) is a quasiconvex function of the weights [8]. Thus the estimation of the running time reduces to choosing the weights minimizing c 1+α 3 .…”

Section: Factmentioning

confidence: 97%

“…The fastest known algorithm for this problem, which dates back to the sixties [20], is based on dynamic programming and has running time (2 n ). Better results are known only for special graph classes, such as cubic graphs [7]. For many other non-local problems the current best known algorithms are still trivial.…”

Section: Introductionmentioning

confidence: 98%

Solving Connected Dominating Set Faster than 2 n

Fomin

Grandoni

Kratsch³

2007

Algorithmica

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In the connected dominating set problem we are given an n-node undirected graph, and we are asked to find a minimum cardinality connected subset S of nodes such that each node not in S is adjacent to some node in S. This problem is also equivalent to finding a spanning tree with maximum number of leaves.Despite its relevance in applications, the best known exact algorithm for the problem is the trivial (2 n ) algorithm that enumerates all the subsets of nodes. This is not the case for the general (unconnected) version of the problem, for which much faster algorithms are available. Such a difference is not surprising, since connectivity is a global property, and non-local problems are typically much harder to solve exactly.In this paper we break the 2 n barrier, by presenting a simple O(1.9407 n ) algorithm for the connected dominating set problem. The algorithm makes use of new domination rules, and its analysis is based on the Measure and Conquer technique.

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“…(The O * -notation means that factors which are polynomial in |G| are suppressed.) The time complexity can be improved to O * (b |G| ), for various constants b with 1 < b < 2, when G has bounded maximum degree, see [6,19,28]. It is conceivable that these algorithms could be modified to give similar results for constructing longest cycles but the only specific results we know of are an algorithm of Monien [37], subsequently improved by Bodlaender [7] to find a longest cycle in an arbitrary graph G in time O (c(G)!…”

Section: Algorithmic Considerationsmentioning

confidence: 99%

Circumference of 3-connected claw-free graphs and large Eulerian subgraphs of 3-edge-connected graphs

Bilinski

Jackson

et al. 2011

Journal of Combinatorial Theory, Series B

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The circumference of a graph is the length of its longest cycles. Results of Jackson, and Jackson and Wormald, imply that the circumference of a 3-connected cubic n-vertex graph is Ω(n 0.694 ), and the circumference of a 3-connected claw-free graph is Ω(n 0.121 ).We generalize and improve the first result by showing that every 3-edge-connected graph with m edges has an Eulerian subgraph with Ω(m 0.753 ) edges. We use this result together with the Ryjáček closure operation to improve the lower bound on the circumference of a 3-connected claw-free graph to Ω(n 0.753 ). Our proofs imply polynomial time algorithms for finding large Eulerian subgraphs of 3-edge-connected graphs and long cycles in 3-connected claw-free graphs.

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The Traveling Salesman Problem for Cubic Graphs

Cited by 84 publications

References 10 publications

Determinant Sums for Undirected Hamiltonicity

Determinant Sums for Undirected Hamiltonicity

Solving Connected Dominating Set Faster than 2 n

Circumference of 3-connected claw-free graphs and large Eulerian subgraphs of 3-edge-connected graphs

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