On the Held-Karp relaxation for the asymmetric and symmetric traveling salesman problems

Authors:
Robert Carr;Santosh Vempala
Affiliations:
Sandia National Labs, Albuquerque, NM 87185, USA. Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy under co ...;Department of Mathematics
Venue:
Mathematical Programming: Series A and B
Year:
2004

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Abstract

A long-standing conjecture in combinatorial optimization says that the integrality gap of the famous Held-Karp relaxation of the metric STSP (Symmetric Traveling Salesman Problem) is precisely 4/3. In this paper, we show that a slight strengthening of this conjecture implies a tight 4/3 integrality gap for a linear programming relaxation of the metric ATSP (Asymmetric Traveling Salesman Problem). Our main tools are a new characterization of the integrality gap for linear objective functions over polyhedra, and the isolation of ‘‘hard-to-round’’ solutions of the relaxations.