An Improved Lower Bound on the Approximability of Metric TSP and Approximation Algorithms for the TSP with Sharpened Triangle Inequality

• Authors: Hans-Joachim Böckenhauer;Juraj Hromkovic;Ralf Klasing;Sebastian Seibert;Walter Unger
• Venue: STACS '00 Proceedings of the 17th Annual Symposium on Theoretical Aspects of Computer Science
• Year: 2000

The traveling salesman problem (TSP) is one of the hardest optimization problems in NPO because it does not admit any polynomial time approximation algorithm (unless P = NP). On the other hand we have a polynomial time approximation scheme (PTAS) for the Euclidean TSP and the 3/2 -approximation algorithm of Christofides for TSP instances satisfying the triangle inequality. The main contributions of this paper are the following: (i) We essentially modify the method of Engebretsen [En99] in order to get a lower bound of 3813/3812-Ɛ on the polynomial-time approximability of the metric TSP for any Ɛ 0. This is an improvement over the lower bound of 5381/5380 -Ɛ in [En99]. Using this approach we moreover prove a lower bound δβ on the approximability of Δβ-TSP for 1/2 β-TSP is a subproblem of the TSP whose input instances satisfy the β-sharpened triangle inequality cost({u, v}) ≤ β ċ (cost({u, v}) + cost({x, v})) for all vertices u, v, x. (ii) We present three different methods for the design of polynomial-time approximation algorithms for Δβ-TSP with 1/2