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


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.


Objective Function Combinatorial Optimization Main Tool Travel Salesman Problem Linear Programming Relaxation 
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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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