Two Approximation Algorithms for ATSP with Strengthened Triangle Inequality

Abstract

In this paper, we study the asymmetric traveling salesman problem (ATSP) with strengthened triangle inequality, i.e. for some $\gamma\in [\frac{1}{2},1)$ 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 $\frac{1}{2(1-\gamma)}$, which is better than all previous results for all $\gamma \in (\frac{1}{2},1)$. The second algorithm is more involved but it also gives a much better approximation ratio: $\frac{2-\gamma}{3(1-\gamma)}+O(\frac{1}{n})$ when γ > γ ₀, and $\frac{1}{2}(1+\gamma)^2 + \epsilon$ for any ε> 0 when γ ≤ γ ₀, where γ ₀ ≈ 0.7003.

This research is partially supported by a grant from the Polish Ministry of Science and Higher Education, project N206 005 32/0807.