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 1, the distances satisfy the inequality dist(x; y) (dist(x; z) + dist(z; y)) for every triple of vertices x, y, and z. We obtain a 4 approximation and also show that for some > 0 it is np-hard to obtain a (1 +) approximation. Our upper bound improves upon the earlier known ratio of (3 2 =2 + =2) 1] for all values of > 7=3.