On The Approximability Of The Traveling Salesman Problem

We show that the traveling salesman problem with triangle inequality cannot be approximated with a ratio better than \( \frac{{117}} {{116}} \) when the edge lengths are allowed to be asymmetric and \( \frac{{220}} {{219}} \) when the edge lengths are symmetric, unless P=NP. The best previous lower bounds were \( \frac{{2805}} {{2804}} \) and \( \frac{{3813}} {{3812}} \) respectively. The reduction is from Håstad’s maximum satisfiability of linear equations modulo 2, and is nonconstructive.

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Correspondence to Christos H. Papadimitriou*.

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* Supported in part by NSF ITR Grant CCR-0121555.

† Supported by NSF award CCR-0307536 and a Sloan foundation fellowship.

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Papadimitriou*, C.H., Vempala†, S. On The Approximability Of The Traveling Salesman Problem. Combinatorica 26, 101–120 (2006). https://doi.org/10.1007/s00493-006-0008-z

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Mathematics Subject Classification (2000):

  • 68Q17
  • 05D40