# Performance Guarantees for the TSP with a Parameterized Triangle Inequality

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## Abstract

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*) ≤τ. (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 *∈* > 0 it is NP-hard to obtain a (1 + *∈τ*) approximation. Our upper bound improves upon the earlier known ratio of (3*τ* ^{2}/2/+*τ*/2)[1] for all values of *τ* > 7/3.

## Keywords

Triangle Inequality Minimum Span Tree Travel Salesman Problem Travel Salesman Problem Hamiltonian Cycle
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