TSP on Cubic and Subcubic Graphs

Boyd, Sylvia C.; Sitters, René; Ster, Suzanne van der; Stougie, Leen

doi:10.1007/978-3-642-20807-2_6

Cited by 23 publications

(47 citation statements)

References 18 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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“…In [14] there is an unproven claim that (1,2)-TSP is APX-hard when the graph of edges of length 1 is cubic, which would imply APXhardness of graph-TSP on cubic graphs. Also note that the 3/2-ratio of Christofides' algorithm is tight for cubic graph-TSP (see [11]). …”

Section: Conjecture 1 For the Metric Tsp The Integrality Gap α(Tsp) mentioning

confidence: 99%

“…Shortly after, improvements for graph-TSP were obtained independently by Aggarwal et al [1], Oveis Gharan et al [30] and by us [10,11]. Aggarwal et al gave a 4/3-approximation for 3-edge-connected cubic graphs by constructing a TSP tour of length at most 4n/3 − 2 when n ≥ 6.…”

Section: Conjecture 1 For the Metric Tsp The Integrality Gap α(Tsp) mentioning

confidence: 99%

“…Independently of all this, we made the next improvement (see Boyd et al [11]) by showing that every bridgeless cubic graph has a TSP tour of length at most 4n/3 − 2 when n ≥ 6. This was the first result which showed that Conjecture 1 is true for graph-TSP, as an upper bound, on cubic bridgeless graphs and it automatically implies a 4/3-approximation algorithm for this class of graph-TSP.…”

Section: Conjecture 1 For the Metric Tsp The Integrality Gap α(Tsp) mentioning

confidence: 99%

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The traveling salesman problem on cubic and subcubic graphs

et al. 2012

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We study the traveling salesman problem (TSP) on the metric completion of cubic and subcubic graphs, which is known to be NP-hard. The problem is of interest because of its relation to the famous 4/3-conjecture for metric TSP, which says that the integrality gap, i.e., the worst case ratio between the optimal value of a TSP instance and that of its linear programming relaxation (the subtour elimination relaxation), is 4/3. We present the first algorithm for cubic graphs with approximation ratio 4/3. The proof uses polyhedral techniques in a surprising way, which is of independent interest. In fact we prove constructively that for any cubic graph on n vertices a tour of length 4n/3 − 2 exists, which also implies the 4/3-conjecture, as an upper bound, that gives a 1.461-approximation for graph-TSP on general graphs and as a side result a 4/3-approximation algorithm for this problem on subcubic graphs, also settling the 4/3-conjecture for this class of graph-TSP. The algorithm by Mömke and Svensson is initially randomized but the authors remark that derandomization is trivial. We will present a different way to derandomize their algorithm which leads to a faster running time. All of the latter also works for multigraphs.

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“…This general approach was also taken by Boyd et al who combined it with polyhedral ideas to obtain approximation guarantees of 4/3 for cubic graphs and 7/5 for subcubic graphs, i.e. graphs with degree at most three at each vertex [BSvdSS11]. Shortly afterwards, Oveis Gharan et al proved that a subtle modification of Christofides' algorithm has an approximation guarantee of 3/2 − ǫ 0 for graph-TSP on general graphs, where ǫ 0 is a fixed constant with value approximately 10 −12 [GSS11].…”

Section: Recent Progress On Graph-tspmentioning

confidence: 99%

An Improved Analysis of the Mömke-Svensson Algorithm for Graph-TSP on Subquartic Graphs

Newman

2014

Lecture Notes in Computer Science

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Mömke and Svensson presented a beautiful new approach for the traveling salesman problem on a graph metric (graph-TSP), which yields a 4/3-approximation guarantee on subcubic graphs as well as a substantial improvement over the 3/2-approximation guarantee of Christofides' algorithm on general graphs. The crux of their approach is to compute an upper bound on the minimum cost of a circulation in a particular network, C(G, T ), where G is the input graph and T is a carefully chosen spanning tree. The cost of this circulation is directly related to the number of edges in a tour output by their algorithm. Mucha subsequently improved the analysis of the circulation cost, proving that Mömke and Svensson's algorithm for graph-TSP has an approximation ratio of at most 13/9 on general graphs. This analysis of the circulation is local, and vertices with degree four and five can contribute the most to its cost. Thus, hypothetically, there could exist a subquartic graph (a graph with degree at most four at each vertex) for which Mucha's analysis of the Mömke-Svensson algorithm is tight. We show that this is not the case and that Mömke and Svensson's algorithm for graph-TSP has an approximation guarantee of at most 25/18 on subquartic graphs. To prove this, we present different methods to upper bound the minimum cost of a circulation on the network C(G, T ). Our approximation guarantee holds for all graphs that have an optimal solution to a standard linear programming relaxation of graph-TSP with subquartic support.

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TSP on Cubic and Subcubic Graphs

Cited by 23 publications

References 18 publications

Short Tours through Large Linear Forests

Short Tours through Large Linear Forests

The traveling salesman problem on cubic and subcubic graphs

An Improved Analysis of the Mömke-Svensson Algorithm for Graph-TSP on Subquartic Graphs

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