Title: Approximating the asymmetric profitable tour

Authors: Viet Hung Nguyen; Thi Thu Thuy Nguyen

Addresses: LIP6, Université Pierre et Marie Curie Paris 6, 4 place Jussieu, 75252, Paris Cedex 05, France. ' Ecole Internationale des Sciences de Traitement de l'Information (EISTI), Avenue du Parc 95011, Cergy Cedex, France

Abstract: We study the version of the asymmetric prize collecting travelling salesman problem, where the objective is to find a directed tour that visits a subset of vertices such that the length of the tour plus the sum of penalties associated with vertices not in the tour is as small as possible. In Dell'Amico et al. (1995), the authors defined it as the Profitable Tour Problem (PTP). We present an (1 + ⌈log(n)⌉)-approximation algorithm for the asymmetric PTP with n is the vertex number. The algorithm that is based on Frieze et al.'s heuristic for the asymmetric travelling salesman problem as well as a method to round fractional solutions of a linear programming relaxation to integers (feasible solution for the original problem), represents a directed version of the Bienstock et al.'s (1993) algorithm for the symmetric PTP.

Keywords: approximation algorithms; asymmetric profitable tour; asymmetric prize collecting TSP; travelling salesman problem; Held-Karp relaxation; discrete optimisation; operations research.

DOI: 10.1504/IJMOR.2012.046689

International Journal of Mathematics in Operational Research, 2012 Vol.4 No.3, pp.294 - 301

Published online: 02 May 2012 *

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