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

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 < 1, where Δβ-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 < β < 1, where the approximation ratio lies between 1 and 3/2, depending on β.