Analysis of a near-metric TSP approximation algorithm

The traveling salesman problem (TSP) is one of the most fundamental optimization problems. We consider the β-metric traveling salesman problem (Δβ-TSP), i.e., the TSP restricted to graphs satisfying the β-triangle inequality c({v,w}) ≤ β(c({v,u}) + c({u,w})), for some cost function c and any three vertices u,v,w. The well-known path matching Christofides algorithm (PMCA) guarantees an approximation ratio of 3β2/2 and is the best known algorithm for the Δβ-TSP, for 1 ≤ β ≤ 2. We provide a complete analysis of the algorithm. First, we correct an error in the original implementation that may produce an invalid solution. Using a worst-case example, we then show that the algorithm cannot guarantee a better approximation ratio. The example can also be used for the PMCA variants for the Hamiltonian path problem with zero and one prespecified endpoints. For two prespecified endpoints, we cannot reuse the example, but we construct another worst-case example to show the optimality of the analysis also in this case.