Approximations for ATSP with Parametrized Triangle Inequality

Abstract

We give a constant factor $(\frac{\gamma } {{1 - \gamma }} + \gamma )$ approximation for the asymmetric traveling salesman problem in graphs with costs on the edges satisfying γ-parametrized triangle inequality (γ-Asymmetric graphs) for γ ∈ [ 1/2 , 1). We also give an improvement of the algorithm with approximation factor approaching $\frac{\gamma } {{1 - \gamma }}$.

We also explore the $\frac{{c_{max} }} {{c_{min} }}$ ratio of edge costs in a general asymmetric graph. We show that for γ ∈ $[\frac{1} {2},\frac{1} {{\sqrt 3 }}), \frac{{c_{max} }} {{c_{min} }} \leqslant \frac{{2\gamma ^3 }} {{1 - 3\gamma ^2 }}$ , while for γ ∈ $[\frac{1} {{\sqrt 3 }},1)$, this ratio can be arbitrarily large. We make use of this result to give a better analysis to our main algorithm. We also observe that when $\frac{{c_{max} }} {{c_{min} }} > \frac{{\gamma ^2 }} {{1 - \gamma - \gamma ^2 }}$ with γ ∈ $(\frac{1} {2},\frac{{\sqrt 5 - 1}} {2})$, the minimum cost and the maximum cost edges in the graph are uniqueand are reverse to each other.

Dept. of Computer Science and Automation, Indian Institute of Science, Bangalore, India 560012. This research is partially supported by Infosys fellowship