An improved approximation algorithm for the ATSP with parameterized triangle inequality

Abstract

In this paper, we consider the asymmetric traveling salesman problem with the γ-parameterized triangle inequality for γ ∈ [ 1 2 , 1 ) . That means the edge weights in the given complete graph G = ( V , E , ω ) satisfy ω ( u , w ) ⩽ γ ⋅ ( ω ( u , v ) + ω ( v , w ) ) for all distinct nodes u , v , w ∈ V . L.S. Chandran and L.S. Ram gave the first constant factor approximation algorithm with polynomial running time for this problem. They achieve performance ratio γ 1 − γ . M. Bläser, B. Manthey and J. Sgall obtain a 1 + γ 2 − γ − γ 3 -approximation algorithm. We devise an approximation algorithm with performance ratio max { 1 + γ 3 1 − γ 2 , γ + γ 2 + 1 2 + γ 3 1 − γ 2 } , which is better than both 1 + γ 2 − γ − γ 3 and γ 1 − γ for almost all γ ∈ [ 1 2 , 1 ) .