A simple LP relaxation for the asymmetric traveling salesman problem
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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.
KeywordsSubtour elimination linear program relaxation Extreme solutions Traveling salesman problem
Mathematics Subject Classification90C27
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- 1.Asadpour, A., Goemans, M.X., Madry, A., Gharan, S.O., Saberi, A.: An o(log n/ log log n)-approximation algorithm for the asymmetric traveling salesman problem. In: SODA, pp. 379–389 (2010)Google Scholar
- 9.Hoffman, A.: Some recent applications of the theory of linear inequalities to extremal combinatorial analysis. In: Proceeding of Symposium in Applied Mathematics, Amer. Math. Soc. pp. 113–127 (1960)Google Scholar
- 12.Lawler E.: The Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization. Wiley-Interscience Series in Discrete Mathematics and Optimization. Wiley, New York (1985)Google Scholar
- 13.Vempala, S., Yannakakis, M.: A convex relaxation for the asymmetric tsp. In: SODA, pp. 975–976 (1999)Google Scholar