Integer Programming and Combinatorial Optimization
Volume 5035 of the series Lecture Notes in Computer Science pp 331343
A Constant Approximation Algorithm for the a priori Traveling Salesman Problem
 David ShmoysAffiliated withCornell University
 , Kunal TalwarAffiliated withMicrosoft Research
Abstract
One of the interesting recent developments in the design and analysis of approximation algorithms has been in the area of algorithms for discrete stochastic optimization problems. In this domain, one is given not just one input, but rather a probability distribution over inputs, and yet the aim is to design an algorithm that has provably good worstcase performance, that is, for any probability distribution over inputs, the objective function value of the solution found by the algorithm must be within a specified factor of the optimal value.
The a priori traveling salesman problem is a prime example of such a stochastic optimization problem. One starts with the standard traveling salesman problem (in which one wishes to find the shortest tour through a given set of points N), and then considers the possibility that only a subset A of the entire set of points is active. The active set is given probabilistically; that is, there is a probability distribution over the subsets of N, which is given as part of the input. The aim is still to compute a tour through all points in N, but in order to evaluate its cost, we instead compute the expectation of the length of this tour after shortcutting it to include only those points in the active set A (where the expectation is computed with respect to the given probability distribution). The goal is to compute a “master tour” for which this expectation is minimized. This problem was introduced in the doctoral theses of Jaillet and Bertsimas, who gave asymptotic analyses when the distances between points in the input set are also given probabilistically.
In this paper, we restrict attention to the socalled “independent activation” model in which we assume that each point j is active with a given probability p _{ j }, and that these independent random events. For this setting, we give a 8approximation algorithm, a polynomialtime algorithm that computes a tour whose a priori TSP objective function value is guaranteed to be within a factor of 8 of optimal (and a randomized 4approximation algorithm, which produces a tour of expected cost within a factor of 4 of optimal). This is the first constant approximation algorithm for this model.
 Title
 A Constant Approximation Algorithm for the a priori Traveling Salesman Problem
 Book Title
 Integer Programming and Combinatorial Optimization
 Book Subtitle
 13th International Conference, IPCO 2008 Bertinoro, Italy, May 2628, 2008 Proceedings
 Pages
 pp 331343
 Copyright
 2008
 DOI
 10.1007/9783540688914_23
 Print ISBN
 9783540688860
 Online ISBN
 9783540688914
 Series Title
 Lecture Notes in Computer Science
 Series Volume
 5035
 Series ISSN
 03029743
 Publisher
 Springer Berlin Heidelberg
 Copyright Holder
 SpringerVerlag Berlin Heidelberg
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 Editors
 Authors

 David Shmoys ^{(1)}
 Kunal Talwar ^{(2)}
 Author Affiliations

 1. Cornell University, Ithaca, NY 14853
 2. Microsoft Research, Silicon Valley Campus, 1065 L’Avenida, Mountain View, CA 94043
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