$\frac{13}{9}$ -Approximation for Graphic TSP

Mucha, Marcin

doi:10.1007/s00224-012-9439-7

Cited by 66 publications

(53 citation statements)

References 9 publications

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“…Mucha [19] gives an improved analysis of the 1.5858-approximation algorithm of Mömke and Svensson [18]; following is from [19]. Proof.…”

Section: Unit-weight Graphical Metricsmentioning

confidence: 99%

“…Proposed by Held and Karp [16] originally for the circuit problem, the Held-Karp relaxation [16] is a standard LP relaxation to (the variants of) TSP, and has been successfully used by many algorithms [6,12,4,2,20,18,19]. In the LP-based design of an approximation algorithm, one important measure of the strength of a particular LP relaxation is its integrality gap, i.e., the worst-case ratio between the integral and fractional optimal values; however, there exists a significant gap between currently known lower and upper bounds on the integrality gap of the Held-Karp relaxation.…”

Section: Introductionmentioning

confidence: 99%

“…In Appendix A, we show how the results of Oveis Gharan et al [20] extend to the path case. Most recently, Mucha [19] gave an improved analysis of Mömke & Svensson's algorithm [18] to prove the performance guarantee of 13/9 for the circuit case and 19/12 + ǫ for the path case, for any ǫ > 0. We observe that the critical case of this analysis is when the Held-Karp optimum is small, and we show how to obtain an algorithm that yields a better performance guarantee on this critical case, based on the main results of this paper.…”

Section: Introductionmentioning

confidence: 99%

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Improving Christofides' Algorithm for the s-t Path TSP

Kleinberg

Shmoys

2015

J. ACM

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We present a deterministic 1+√ 5 2 -approximation algorithm for the s-t path TSP for an arbitrary metric. Given a symmetric metric cost on n vertices including two prespecified endpoints, the problem is to find a shortest Hamiltonian path between the two endpoints; Hoogeveen showed that the natural variant of Christofides' algorithm is a 5/3-approximation algorithm for this problem, and this asymptotically tight bound in fact has been the best approximation ratio known until now. We modify this algorithm so that it chooses the initial spanning tree based on an optimal solution to the Held-Karp relaxation rather than a minimum spanning tree; we prove this simple but crucial modification leads to an improved approximation ratio, surpassing the 20-year-old barrier set by the natural Christofides' algorithm variant. Our algorithm also proves an upper bound ofon the integrality gap of the pathvariant Held-Karp relaxation. The techniques devised in this paper can be applied to other optimization problems as well: these applications include improved approximation algorithms and improved LP integrality gap upper bounds for the prize-collecting s-t path problem and the unit-weight graphical metric s-t path TSP. *

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“…Mucha [19] gives an improved analysis of the 1.5858-approximation algorithm of Mömke and Svensson [18]; following is from [19]. Proof.…”

Section: Unit-weight Graphical Metricsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Improving Christofides' Algorithm for the s-t Path TSP

Kleinberg

Shmoys

2015

J. ACM

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“…Gharan, Saberi and Singh [11] gave an (1.5 − )-approximation algorithm, for of the order 10 −12 . Mömke and Svensson [21] suggested an algorithm with a better approximation of 1.461, which was later improved to 13/9 by a better analysis of Mucha [22]. The best known approximation algorithm for the graphic TSP is 7/5 due to Sebö and Vygen [30].…”

Section: Introductionmentioning

confidence: 99%

An Improved Upper Bound for the Universal TSP on the Grid

Christodoulou¹,

Sgouritsa²

2017

Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms

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We study the universal Traveling Salesman Problem in an n × n grid with the shortest path metric. The goal is to define a (universal) total ordering over the set of grid's vertices, in a way that for any input (subset of vertices), the tour, which visits the points in this ordering, is a good approximation of the optimal tour, i.e. has low competitive ratio.This problem was first studied by Platzman and Bartholdi [26]. They proposed a heuristic, which was based on the Sierpinski space-filling curve, in order to define a universal ordering of the unit square [0, 1] 2 under the Euclidean metric. Their heuristic visits the points of the unit square in the order of their appearance along the space-filling curve. They provided a logarithmic upper bound which was shown to be tight up to a constant by Bertsimas and Grigni [3]. Bertsimas and Grigni further showed logarithmic lower bounds for other space-filling curves and they conjectured that any universal ordering has a logarithmic lower bound for the n × n grid.In this work, we disprove this conjecture by showing that there exists a universal ordering of the n × n grid with competitive ratio of O log n log log n . The heuristic we propose defines a universal ordering of the grid's vertices based on a generalization of the Lebesgue spacefilling curve. In order to analyze the competitive ratio of our heuristic, we employ techniques from the theory of geometric spanners in Euclidean spaces. We finally show that our analysis is tight up to a constant.

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“…The difficulty of the exploration problem in general dynamic graphs is further underlined by the fact that the exploration problem for static graphs is the well-known Graph TSP problem (see e.g. [8,9,11]), which is already APX hard in general graphs.…”

Section: Introductionmentioning

confidence: 99%

Exploration of Constantly Connected Dynamic Graphs Based on Cactuses

Ilcinkas

Klasing

Wade

2014

Lecture Notes in Computer Science

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Abstract. We study the problem of exploration by a mobile entity (agent) of a class of dynamic networks, namely constantly connected dynamic graphs. This problem has already been studied in the case where the agent knows the dynamics of the graph and the underlying graph is a ring of n vertices [5]. In this paper, we consider the same problem and we suppose that the underlying graph is a cactus graph (a connected graph in which any two simple cycles have at most one vertex in common). We propose an algorithm that allows the agent to explore these dynamic graphs in at most 2 O( √ log n) n time units. We show that the lower bound of the algorithm is 2 Ω( √ log n) n time units.

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$\frac{13}{9}$ -Approximation for Graphic TSP

Cited by 66 publications

References 9 publications

Improving Christofides' Algorithm for the s-t Path TSP

Improving Christofides' Algorithm for the s-t Path TSP

An Improved Upper Bound for the Universal TSP on the Grid

Exploration of Constantly Connected Dynamic Graphs Based on Cactuses

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