Elsevier

Discrete Optimization

Volume 8, Issue 4, November 2011, Pages 525-539
Discrete Optimization

Finding low cost TSP and 2-matching solutions using certain half-integer subtour vertices

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Abstract

Consider the traveling salesman problem (TSP) defined on the complete graph, where the edge costs satisfy the triangle inequality. Let TOUR denote the optimal solution value for the TSP. Two well-known relaxations of the TSP are the subtour elimination problem and the 2-matching problem. If we let SUBT and 2M represent the optimal solution values for these two relaxations, then it has been conjectured that TOUR/SUBT 4/3, and that 2M/SUBT 10/9.

In this paper, we exploit the structure of certain 1/2-integer solutions for the subtour elimination problem in order to obtain low cost TSP and 2-matching solutions. In particular, we show that for cost functions for which the optimal subtour elimination solution found falls into our classes, the above two conjectures are true. Our proofs are constructive and could be implemented in polynomial time, and thus, for such cost functions, provide a 4/3 (or better) approximation algorithm for the TSP.

Keywords

Traveling salesman problem
2-matching
Approximation algorithm

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This research was partially supported by grants from the Natural Sciences and Engineering Research Council of Canada.

1

Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy’s National Nuclear Security Administration under contract DE-AC04-94AL85000.