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Mathematical Programming

, Volume 100, Issue 3, pp 569–587 | Cite as

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

  • Robert CarrEmail author
  • Santosh Vempala
Article

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.

Keywords

Objective Function Combinatorial Optimization Main Tool Travel Salesman Problem Linear Programming Relaxation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  1. 1.Sandia National LabsUSA. Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy under contract DE-AC04-94AL85000
  2. 2.Department of Mathematics

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