TSP Tours in Cubic Graphs: Beyond 4/3

Correa, José R.; Larré, Omar; Soto, José A.

doi:10.1007/978-3-642-33090-2_68

Cited by 7 publications

(8 citation statements)

References 13 publications

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“…While Christofides' algorithm [6] gives a 3 2 approximation even for graph-TSP, a small but constant improvement was presented by Oveis-Gharan et al [17]. Mömke and Svensson [14] improved this significantly while further improvements by Sebö and Vygen [18] have brought the current best approximation factor for graph-TSP down to Another line of work has focused on graph theoretic methods to obtain improved approximation factors: Boyd et al [5] showed a 4 3 approximation for 2-connected cubic graphs; Correa et al [7] gave an algorithm that finds a tour of length at most ( )n in n-node 2-connected cubic graphs, while Karp and Ravi [12] gave an algorithm that finds a tour of length at most 9n 7 in cubic bipartite graphs. For general, d-regular connected graphs, Vishnoi [19] gave an algorithm for finding tours of length at most…”

Section: Related Workmentioning

confidence: 99%

Short Tours through Large Linear Forests

Feige

Ravi

Singh

2014

Integer Programming and Combinatorial Optimization

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Abstract. A tour in a graph is a connected walk that visits every vertex at least once, and returns to the starting vertex. Vishnoi (2012) proved that every connected d-regular graph with n vertices has a tour of length at most (1 + o (1))n, where the o(1) term (slowly) tends to 0 as d grows. His proof is based on van-derWarden's conjecture (proved independently by Egorychev (1981) and by Falikman (1981)) regarding the permanent of doubly stochastic matrices. We provide an exponential improvement in the rate of decrease of the o(1) term (thus increasing the range of d for which the upper bound on the tour length is nontrivial). Our proof does not use the van-der-Warden conjecture, and instead is related to the linear arboricity conjecture of Akiyama, Exoo and Harary (1981), or alternatively, to a conjecture of Magnant and Martin (2009) regarding the path cover number of regular graphs. More generally, for arbitrary connected graphs, our techniques provide an upper bound on the minimum tour length, expressed as a function of their maximum, average, and minimum degrees. Our bound is best possible up to a term that tends to 0 as the minimum degree grows.

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Section: Related Workmentioning

confidence: 99%

Short Tours through Large Linear Forests

Feige

Ravi

Singh

2014

Integer Programming and Combinatorial Optimization

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“…at most 4n/3 in a 3-edge-connected cubic graph of order n. This result was further improved to 2-edge connected or connected cubic graphs, graphs of maximum degree at most 3, or better bounds than 4n/3; see [6,8,9]. Because of the above reasons, several researchers have been interested in a 2-factor in cubic graphs such that the number of 5-cycles is small; see [7].…”

Section: Every Vertex Inmentioning

confidence: 95%

“…Instead of using a 2-factor, we can use an even subgraph satisfying certain conditions on the order of each component. In fact, such structures have appeared in [8,9] as intermediate products, which is called an R-factor in [8]. For those intermediate products, it is not necessarily dominating, but the dominating property may help us to obtain good bounds, i.e., we expect that Theorem 1.1 has a potential application to the TSP.…”

Section: Every Vertex Inmentioning

confidence: 99%

On Dominating Even Subgraphs in Cubic Graphs

Čada¹,

Chiba²,

Ozeki³

et al. 2017

SIAM J. Discrete Math.

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It is known that a 3-edge-connected graph has a spanning even subgraph in which every component contains at least five vertices, and the lower bound is best possible. A natural question arises of whether we can improve the lower bound by changing the spanning property with the dominating property. In this paper, we show that a 3-edge-connected cubic graph has a dominating even subgraph in which every component contains at least six vertices.

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“…Boyd et al [1] proved that a simple 2-connected cubic graph on n vertices has a TSP tour of length at most 4/3 • n − 2 (assuming n ≥ 6). The result was subsequently improved by Correa, Larré, and Soto [3] to (4/3 − 1/61236) • n, by van Zuylen [10] to (4/3 − 1/8754) • n, by Candráková and Lukoťka [2] to 1.3 • n, and very recently by Dvořák, Kráľ and Mohar [4] to 9/7 • n.…”

Section: Traveling Salesman Problem In Unweighted Graphsmentioning

confidence: 99%

Simple cubic graphs with no short traveling salesman tour

Lukoťka

Mazák

2018

Discrete Applied Mathematics

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Let tsp(G) denote the length of a shortest travelling salesman tour in a graph G. We prove that for any ε > 0, there exists a simple 2-connected planar cubic graph G 1 such that tsp(G 1 ) ≥ (1.25 − ε) · |V (G 1 )|, a simple 2-connected bipartite cubic graph G 2 such that tsp(G 2 ) ≥ (1.2 − ε) · |V (G 2 )|, and a simple 3-connected cubic graph G 3 such that tsp(G 3 ) ≥ (1.125 − ε) · |V (G 3 )|.

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TSP Tours in Cubic Graphs: Beyond 4/3

Cited by 7 publications

References 13 publications

Short Tours through Large Linear Forests

Short Tours through Large Linear Forests

On Dominating Even Subgraphs in Cubic Graphs

Simple cubic graphs with no short traveling salesman tour

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