Analysis of approximation algorithms for the traveling salesman problem in near-metric graphs

We consider the beta-metric traveling salesman problem (Delta-beta-TSP), i.e., the TSP restricted to input instances satisfying the beta-triangle inequality c({v,w}) <= beta * (c{v,u} + c{u,w}), for any three vertices u,v,w. The well-known path matching Christofides algorithm (PMCA) provides an approximation ratio of 3/2 * beta^2 and is the best known algorithm in the range 1 <= beta <= 2. We show that this upper bound is tight by providing a worst-case example. This example can also be used to show the tightness of the upper bound 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 tightness of the upper bound also in this case. Furthermore, we establish improved lower bounds for an approximation algorithm for the metric Hamiltonian path problem as well as for two approximation algorithms for the metric TSP reoptimization problem.