Performance guarantees for the TSP with a parameterized triangle inequality

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 e 0 >0 it is NP-hard to obtain a (1+e 0 τ) -approximation for all τ≥1 . Our upper bound improves upon the earlier known ratio of (τ 2 +τ) [? ] for all values of τ>3 .