Two Approximation Algorithms for ATSP with Strengthened Triangle Inequality

In this paper, we study the asymmetric traveling salesman problem (ATSP) with strengthened triangle inequality, i.e. for some γ ∈ [ 1 2 , 1) the edge weights satisfy w(u, v) ≤ γ(w(u, x) + w(x, v)) for all distinct vertices u, v, x. We present two approximation algorithms for this problem. The first one is very simple and has approximation ratio 1 2(1−γ) , which is better than all previous results for all γ ∈ ( 1 2 , 1). The second algorithm is more involved but it also gives a much better approximation ratio: 2−γ 3(1−γ) + O( 1 n ) when γ > γ0, and 1 2 (1 + γ) + for any > 0 when γ ≤ γ0, where γ0 ≈ 0.7003.