Performance guarantees for the TSP with a parameterized triangle inequality

We consider the approximability of the tsp problem in graphs that satisfy a relaxed form of triangle inequality. More precisely, we assume that for some parameter Τ ≥ 1, the distances satisfy the inequality dist(x, y) ≤ Τċ(dist(x, z)+dist(z, y)) for every triple of vertices x, y, and z. We obtain a 4Τ approximation and also show that for some Ɛ > 0 it is np-hard to obtain a (1+ ƐΤ) approximation. Our upper bound improves upon the earlier known ratio of (3Τ2=2+Τ2) [1] for all values of Τ < 7=3.

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