Approximation Algorithms with Bounded Performance Guarantees for the Clustered Traveling Salesman Problem

Guttmann‐Beck, Nili; Hassin, Refael; Khuller, Samir; Raghavachari, Balaji

doi:10.1007/978-3-540-49382-2_2

Cited by 26 publications

(40 citation statements)

References 12 publications

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“…We show for instance that in any worst case instance for Hoogeveen's algorithm, the minimum spanning tree involved in the algorithm contains a path between the end vertices of cost exactly 1/3 of the cost of an optimal solution and a gradual relaxation of this bound for inputs that do not cause worst-case behavior. This generalizes some of the results from [15]. The properties revealed in this work restrict the types of inputs that a possible improved algorithm for the ∆HPP 2 has to cope with.…”

Section: Introductionsupporting

confidence: 81%

“…The situation is very similar for the metric Hamiltonian path problem with prespecified start and end vertex (∆HPP 2 ): the lower bound is closely related to that of the ∆TSP and the 5/3-approximative algorithm by Hoogeveen [16] was not improved so far. An alternative proof for the same result was given by Guttmann-Beck et al [15]. For both problems, the upper bounds on the approximability have resisted all attempts of improvement for many years.…”

Section: Introductionmentioning

confidence: 70%

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Structural Properties of Hard Metric TSP Inputs

Mömke

2011

SOFSEM 2011: Theory and Practice of Computer Science

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The metric traveling salesman problem is one of the most prominent APX-complete optimization problems. An important particularity of this problem is that there is a large gap between the known upper bound and lower bound on the approximability (assuming P = N P ). In fact, despite more than 30 years of research, no one could find a better approximation algorithm than the 1.5-approximation provided by Christofides. The situation is similar for a related problem, the metric Hamiltonian path problem, where the start and the end of the path are prespecified: the best approximation ratio up to date is 5/3 by an algorithm presented by Hoogeveen almost 20 years ago.In this paper, we provide a tight analysis of the combined outcome of both algorithms. This analysis reveals that the sets of the hardest input instances of both problems are disjoint in the sense that any input is guaranteed to allow at least one of the two algorithms to achieve a significantly improved approximation ratio. In particular, we show that any input instance that leads to a 5/3-approximation with Hoogeveen's algorithm enables us to find an optimal solution for the traveling salesman problem. This way, we determine a set S of possible pairs of approximation ratios. Furthermore, for any input we can identify one pair of approximation ratios within S that forms an upper bound on the achieved approximation ratios.

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Section: Introductionsupporting

confidence: 81%

Section: Introductionmentioning

confidence: 70%

Structural Properties of Hard Metric TSP Inputs

Mömke

2011

SOFSEM 2011: Theory and Practice of Computer Science

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“…Consider the following algorithm: For every pairf ,f of disjoint edges, both of which are adjacent to e, compute an approximate solution to the TSP path problem on the subgraph of G N induced by the vertex set V \ e (i. e., without two vertices) with start vertexṽ and end vertexṽ where {ṽ} =f \ e and {ṽ } =f \ e. It is known [13,14] that this can be done with an approximation guarantee of 5 3 . Each of these paths is augmented byf , e, andf so as to yield a TSP tour.…”

Section: The Metric Casementioning

confidence: 99%

Reusing Optimal TSP Solutions for Locally Modified Input Instances

Böckenhauer

Forlizzi

Hromkovič

et al.

Fourth IFIP International Conference on Theoretical Computer Science- TCS 2006

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Abstract. Given an instance of an optimization problem together with an optimal solution, we consider the scenario in which this instance is modified locally. In graph problems, e. g., a singular edge might be removed or added, or an edge weight might be varied, etc. For a problem U and such a local modification operation, let lm-U (local-modification-U ) denote the resulting problem. The question is whether it is possible to exploit the additional knowledge of an optimal solution to the original instance or not, i. e., whether lm-U is computationally more tractable than U . Here, we give non-trivial examples both of problems where this is and problems where this is not the case. Our main results are these:1. The local modification to change the cost of a singular edge turns the traveling salesperson problem (TSP) into a problem lm-TSP which is as hard as TSP itself, i. e., unless P = N P , there is no polynomial-time p(n)-approximation algorithm for lm-TSP for any polynomial p. Moreover, lm-TSP where inputs must satisfy the β-triangle inequality (lm-∆ β -TSP) remains NP-hard for all β > 1 2 . 2. For lm-∆-TSP (i. e., metric lm-TSP), an efficient 1.4-approximation algorithm is presented. In other words, the additional information enables us to do better than if we simply used Christofides' algorithm for the modified input. 3. Similarly, for all 1 < β < 3.34899, we achieve a better approximation ratio for lm-∆ β -TSP than for ∆ β -TSP. 4. Metric TSP with deadlines (time windows), if a single deadline or the cost of a single edge is modified, exhibits the same lower bounds on the approximability in these local-modification versions as those currently known for the original problem.

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“…No caso particular do PCVG ter somente um grupo ou analogamente um vértice em cada grupo, o PCVG torna-se PCV. Neste sentido, o PCVGé também N P-difícil [6]. Uma descrição do PCVG na estrutura de um grafoé dado a seguir: seja G=(V, A) um grafo completo, direcionado, simétrico e ponderado com um conjunto de vértices V = {v 1 , v 2 , ..., v n } e um conjunto de arcos …”

Section: Definição Do Problemaunclassified

“…A escolha da ordem de visita de cada grupo tambémé aleatória. Os algoritmos α-aproximados para o PCVG com diferentes variantes são encontrados em [1] e [6]. A maioria destes algoritmos necessita estabelecer em cada grupo os vértices de entrada e de saída e uma ordem pré-definida de visita dos grupos.…”

Section: Introductionunclassified

Um Método Exato para o Problema do Caixeiro Viajante com Grupamentos Euclidiano e Simétrico

Mestria

2015

Proceeding Series of the Brazilian Society of Computational and Applied Mathematics

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Resumo: Nesse artigo,é proposto um método exato para resolver o Problema do Caixeiro Viajante com Grupamentos (PCVG). O PCVGé uma generalização do Problema do Caixeiro Viajante (PCV), onde os vértices são particionados em grupos disjuntos e o objetivoé encontrar um ciclo hamiltoniano de custo mínimo tal que os vértices de cada grupo são visitados de forma contígua. A formulação para o método inclui a restrição de Chisman (1975)

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Approximation Algorithms with Bounded Performance Guarantees for the Clustered Traveling Salesman Problem

Cited by 26 publications

References 12 publications

Structural Properties of Hard Metric TSP Inputs

Structural Properties of Hard Metric TSP Inputs

Reusing Optimal TSP Solutions for Locally Modified Input Instances

Um Método Exato para o Problema do Caixeiro Viajante com Grupamentos Euclidiano e Simétrico

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