Approximation algorithms for the TSP with sharpened triangle inequality

Abstract 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. In this paper, we consider Δ β -TSP, for 1/2 , as a subproblem of the TSP whose input instances satisfy the β -sharpened triangle inequality cost({u,v})≤β(cost({u,x})+cost({x,v})) for all vertices u , v , x . This problem is APX-complete for every β>1/2 . The main contribution of this paper is the presentation of three different methods for the design of polynomial-time approximation algorithms for Δ β -TSP with 1/2 , where the approximation ratio lies between 1 and 3/2 , depending on β .