Approximation algorithms for the 2-peripatetic salesman problem with edge weights 1 and 2

Baburin, A. E.; Croce, Federico Della; Гимади, Э. Х.; Glazkov, Yu. V.; Paschos, Vangélis Th.

doi:10.1016/j.dam.2008.06.025

Cited by 17 publications

(6 citation statements)

References 9 publications

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“…The purpose of this work is to strengthen the results of [4], namely, to construct an approximation algorithm for solving the problem 2-PSP(1,2)-min-2w with the approximation ratio 4/3 (without any additive constant) and an estimation of the time complexity O(n 5 ). The algorithm is based on the idea of the method of [5] consisting in construction and sequential "improvement" of a couple of edge-disjoint partial tours (sets of chains and cycles) of edges with unit weight and the subsequent closure of these tours into the disjoint Hamiltonian cycles.…”

Section: Introductionmentioning

confidence: 97%

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An approximation algorithm for the minimum two peripatetic salesmen problem with different weight functions

Glebov

Zambalaeva

2012

J. Appl. Ind. Math.

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Section: Introductionmentioning

confidence: 97%

“…Gimadi, and A. N. Glebov proposed an algorithm with ratio 6/5 in the case when the weight functions for both cycles are identical [2]. In the case of the two different weight functions (Problem 2-PSP(1,2)-min-2w), an algorithm with ratio 11/7 is proposed in [4].…”

Section: Introductionmentioning

confidence: 99%

An approximation algorithm for the minimum two peripatetic salesmen problem with different weight functions

Glebov

Zambalaeva

2012

J. Appl. Ind. Math.

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“…Alfandari and Paschos [59] modeled a real-case network design problem facing by a French telecommunications company as the 2H-MSTP and showed that it cannot be approximated with a ratio better than O(log n). The famous Traveling Salesperson problem and Steiner Tree problem have been studied extensively under the (1, 2)-setting such that each edge in the considered graph has weight 1 or 2 [60,61,62,63,64] (one can consider that the weights on the edges are not well-defined, just "small" and "large"). Thus Alfandari and Paschos [59] also investigated the 2H-MSTP under the (1, 2)-setting and gave an approximation algorithm with ratio 5/4.…”

Section: Introductionmentioning

confidence: 99%

Time Complexity Analysis of Evolutionary Algorithms for 2-Hop (1,2)-Minimum Spanning Tree Problem

Shi¹,

Neumann²,

Wang³

2021

Preprint

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The Minimum Spanning Tree problem (abbr. MSTP) is a well-known combinatorial optimization problem that has been extensively studied by the researchers in the field of evolutionary computing to theoretically analyze the optimization performance of evolutionary algorithms. Within the paper, we consider a constrained version of the problem named 2-Hop (1,2)-Minimum Spanning Tree problem (abbr. 2H-(1,2)-MSTP) in the context of evolutionary algorithms, which has been shown to be NP-hard. Following how evolutionary algorithms are applied to solve the MSTP, we first consider the evolutionary algorithms with search points in edge-based representation adapted to the 2H-(1,2)-MSTP (including the (1+1) EA, Global Simple Evolutionary Multi-Objective Optimizer and its two variants). More specifically, we separately investigate the upper bounds on their expected time (i.e., the expected number of fitness evaluations) to obtain a 3 2 -approximate solution with respect to different fitness functions. Inspired by the special structure of 2-hop spanning trees, we also consider the (1+1) EA with search points in vertex-based representation that seems not so natural for the problem and give an upper bound on its expected time to obtain a 3 2 -approximate solution, which is better than the above mentioned ones.

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“…The 2-peripatetic salesperson is NP-complete even for 4-regular graphs [12]. Much attention was paid to the development of approximation algorithms for this problem (see, for example, [1,4,16]).…”

Section: Introductionmentioning

confidence: 99%

Simulated Annealing Approach to Verify Vertex Adjacencies in the Traveling Salesperson Polytope

Kozlova

Nikolaev

2019

Mathematical Optimization Theory and Operations Research

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We consider 1-skeletons of the symmetric and asymmetric traveling salesperson polytopes whose vertices are all possible Hamiltonian tours in the complete directed or undirected graph, and the edges are geometric edges or one-dimensional faces of the polytope. It is known that the question whether two vertices of the symmetric or asymmetric traveling salesperson polytopes are nonadjacent is NP-complete. A sufficient condition for nonadjacency can be formulated as a combinatorial problem: if from the edges of two Hamiltonian tours we can construct two complementary Hamiltonian tours, then the corresponding vertices of the traveling salesperson polytope are not adjacent. We consider a heuristic simulated annealing approach to solve this problem. It is based on finding a vertex-disjoint cycle cover and a perfect matching. The algorithm has a one-sided error: the answer "not adjacent" is always correct, and was tested on random and pyramidal Hamiltonian tours.Keywords: traveling salesperson problem · Hamiltonian tour · traveling salesperson polytope · 1-skeleton · vertex adjacency · simulated annealing · vertex-disjoint cycle cover · perfect matching.

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Approximation algorithms for the 2-peripatetic salesman problem with edge weights 1 and 2

Cited by 17 publications

References 9 publications

An approximation algorithm for the minimum two peripatetic salesmen problem with different weight functions

An approximation algorithm for the minimum two peripatetic salesmen problem with different weight functions

Time Complexity Analysis of Evolutionary Algorithms for 2-Hop (1,2)-Minimum Spanning Tree Problem

Simulated Annealing Approach to Verify Vertex Adjacencies in the Traveling Salesperson Polytope

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