Mathematical Programming

, Volume 141, Issue 1–2, pp 549–559 | Cite as

A simple LP relaxation for the asymmetric traveling salesman problem

Full Length Paper Series A

Abstract

It is a long-standing open question in combinatorial optimization whether the integrality gap of the subtour linear program relaxation (subtour LP) for the asymmetric traveling salesman problem (ATSP) is a constant. The study on the structure of this linear program is important and extensive. In this paper, we give a new and simpler LP relaxation for the ATSP. Our linear program consists of a single type of constraints that combine both the subtour elimination and the degree constraints in the traditional subtour LP. As a result, we obtain a much simpler relaxation. In particular, it is shown that the extreme solutions of our program have at most 2n − 2 non-zero variables, improving the bound 3n − 2, proved by Vempala and Yannakakis, for the ones obtained by the subtour LP. Nevertheless, the integrality gap of the new linear program is larger than that of the traditional subtour LP by at most a constant factor.

Keywords

Subtour elimination linear program relaxation Extreme solutions Traveling salesman problem 

Mathematics Subject Classification

90C27 

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Copyright information

© Springer and Mathematical Optimization Society 2012

Authors and Affiliations

  1. 1.Managerial Economics and Decision Sciences, Kellogg School of ManagementNorthwestern UniversityEvanstonUSA

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