A Simple LP Relaxation for the Asymmetric Traveling Salesman Problem

Authors:
Thành Nguyen
Affiliations:
Center for Applies Mathematics, Cornell University, Ithaca, USA NY 14853
Venue:
APPROX '08 / RANDOM '08 Proceedings of the 11th international workshop, APPROX 2008, and 12th international workshop, RANDOM 2008 on Approximation, Randomization and Combinatorial Optimization: Algorithms and Techniques
Year:
2008

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Abstract

A long-standing conjecture in Combinatorial Optimization is that the integrality gap of the Held-Karp LP relaxation for the Asymmetric Traveling Salesman Problem (ATSP) is a constant. In this paper, we give a simpler LP relaxation for the ASTP. The integrality gaps of this relaxation and of the Held-Karp relaxation are within a constant factor of each other. Our LP is simpler in the sense that its extreme solutions have at most 2n茂戮驴 2 non-zero variables, improving the bound 3n茂戮驴 2 proved by Vempala and Yannakakis for the extreme solutions of the Held-Karp LP relaxation. Moreover, more than half of these non-zero variables can be rounded to integers while the total cost only increases by a constant factor.We also show that given a partially rounded solution, in an extreme solution of the corresponding LP relaxation, at least one positive variable is greater or equal to 1/2.