The Colorful Traveling Salesman Problem

Xiong, Yupei; Golden, Bruce L.; Wasil, Edward

doi:10.1007/978-0-387-48793-9_8

Cited by 23 publications

(7 citation statements)

References 8 publications

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“…The values of d and h are given in ( 17) and (18), which are both functions of k. As for k, we only need it to be a sufficiently large integer. The specific requirement on k (how large k should be) is given in (24).…”

Section: Construction Of the Instancementioning

confidence: 99%

“…Besides the Label s-t Cut problem, there are still many classic optimization problems that have been considered under the edge-classified model, such as the Min Label Spanning Tree problem [5,18], the Min Label s-t Path problem [4,13], the Min Label Traveling Salesman problem [9,24], the Min Label Perfect Matching problem [19], and the Min Label Steiner Tree problem [7], etc.…”

Section: Introductionmentioning

confidence: 99%

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Minimum Label s-t Cut has large integrality gaps

Zhang

Tang

2020

Information and Computation

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Given a graph G = (V, E) with a label set L = { 1 , 2 , . . . , q }, in which each edge has a label from L, a source s ∈ V , and a sink t ∈ V , the Min Label s-t Cut problem asks to pick a set L ⊆ L of labels with minimized cardinality, such that the removal of all edges with labels in L from G disconnects s and t. This problem comes from many applications in real world, for example, information security and computer networks. In this paper, we study two linear programs for Min Label s-t Cut, proving that both of them have large integrality gaps, namely, Ω(m) and Ω(m 1/3− ) for the respective linear programs, where m is the number of edges in the graph and > 0 is any arbitrarily small constant. As Min Label s-t Cut is NP-hard and the linear programming technique is a main approach to design approximation algorithms, our results give negative answer to the hope that designs better approximation algorithms for Min Label s-t Cut that purely rely on linear programming.

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Section: Construction Of the Instancementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Minimum Label s-t Cut has large integrality gaps

Zhang

Tang

2020

Information and Computation

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“…If at most q colors appear in the graph, a 2q approximation algorithm is shown. The MinLTSP has also been experimentally investigated in [19] by Xiong, Golden and Wasil.…”

Section: Related Workmentioning

confidence: 99%

Labeled Traveling Salesman Problems: Complexity and approximation

Couëtoux

Gourvès

Monnot

et al. 2010

Discrete Optimization

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We consider labeled Traveling Salesman Problems, defined upon a complete graph of n vertices with colored edges. The objective is to find a tour of maximum or minimum number of colors. We derive results regarding hardness of approximation and analyze approximation algorithms, for both versions of the problem. For the maximization version we give a 1 2-approximation algorithm based on local improvements and show that the problem is APXhard. For the minimization version, we show that it is not approximable within n 1−ǫ for any fixed ǫ > 0. When every color appears in the graph at most r times and r is an increasing function of n, the problem is shown not to be approximable within factor O(r 1−ǫ). For fixed constant r we analyze a polynomial-time (r + H r)/2-approximation algorithm, where H r is the r-th harmonic number, and prove APX-hardness for r = 2. For all of the analyzed algorithms we exhibit tightness of their analysis by provision of appropriate worst-case instances.

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“…Several other problems defined on graphs with labeled edges have been defined and studied. Among them, we recall the colorful traveling salesman problem , the labeled perfect matching problem and the rainbow spanning forest problem . A survey on edge‐labeled problems, as well as a variant of the maximum flow problem, are presented in .…”

Section: Introductionmentioning

confidence: 99%

Heuristics for the strong generalized minimum label spanning tree problem

2019

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In this work we introduce and study the strong generalized minimum label spanning tree (GMLST), a novel optimization problem defined on edge-labeled graphs. Given a label set associated to each edge of the input graph, the aim is to look for the spanning tree using the minimum number of labels. Differently from the previously introduced GMLST problem, including a given edge in the solution means that all its labels are used. We present a mathematical formulation, as well as three heuristic approaches to solve the problem. Computational results compare the performances of the proposed algorithms. KEYWORDS carousel greedy, generalized problem, minimum label spanning tree, pilot method i=1 1 i , where b is the maximum number of occurrences of a label. The authors also showed this new bound to be tight. Metaheuristics for the problem were proposed in several works [4,7,11,19,22]. In [1], the authors presented a single-commodity flow mathematical model, and showed that the formulation obtained by relaxing integrality on the arc selection variables always provides the optimal solution value for the original unrelaxed problem.Several other problems defined on graphs with labeled edges have been defined and studied. Among them, we recall the colorful traveling salesman problem [6,13,14,17,24], the labeled perfect matching problem [16] and the rainbow spanning forest problem [2]. A survey on edge-labeled problems, as well as a variant of the maximum flow problem, are presented in [12].Networks. 2019;74:148-160.wileyonlinelibrary.com/journal/net

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The Colorful Traveling Salesman Problem

Cited by 23 publications

References 8 publications

Minimum Label s-t Cut has large integrality gaps

Minimum Label s-t Cut has large integrality gaps

Labeled Traveling Salesman Problems: Complexity and approximation

Heuristics for the strong generalized minimum label spanning tree problem

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