Analysis of a near-metric TSP approximation algorithm

Krug, Sacha

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 47 (2013) no. 3, pp. 293-314.

The traveling salesman problem (TSP) is one of the most fundamental optimization problems. We consider the β-metric traveling salesman problem (Δ_β-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 Δ_β-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.

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