Performance guarantees for the TSP with a parameterized triangle inequality
Information Processing Letters
Algorithmics for Hard Problems
Algorithmics for Hard Problems
Discrete Mathematics with Graph Theory (3rd Edition)
Discrete Mathematics with Graph Theory (3rd Edition)
Introduction to Algorithms, Third Edition
Introduction to Algorithms, Third Edition
Structural properties of hard metric TSP inputs
SOFSEM'11 Proceedings of the 37th international conference on Current trends in theory and practice of computer science
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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 $\frac{3}{2}\beta^2$ 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 be reused 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.