Approximating the Metric TSP in Linear Time

Abstract

Given a metric graph G=(V,E) of n vertices, i.e., a complete graph with a non-negative real edge cost function satisfying the triangle inequality, the metricity degree of G is defined as $\beta=\max_{x,y,z\in V}\{\frac{c(x,y)}{c(x,z)+c(y,z)}\}\in[\frac{1}{2},1]$. This value is instrumental to establish the approximability of several NP-hard optimization problems definable on G, like for instance the prominent traveling salesman problem, which asks for finding a Hamiltonian cycle of G of minimum total cost. In fact, this problem can be approximated quite accurately depending on the metricity degree of G, namely by a ratio of either $\frac{2-\beta}{3(1-\beta)}$ or $\frac{3\beta^{2}}{3\beta^{2}-2\beta+1}$, for $\beta<\frac{2}{3}$ or $\beta\geq \frac{2}{3}$, respectively. Nevertheless, these approximation algorithms have O(n ³) and O(n ^2.5log ^1.5 n) running time, respectively, and therefore they are superlinear in the Θ(n ²) input size. Thus, since many real-world problems are modeled by graphs of huge size, their use might turn out to be unfeasible in practice, and alternative approaches requiring only O(n ²) time are sought. However, with this restriction, all the currently available approaches can only guarantee a 2-approximation ratio for the case β=1, which means a $\frac{2\beta^{2}}{2\beta^{2}-2\beta+1}$-approximation ratio for general β<1. In this paper, we show how to elaborate—without affecting the space and time complexity—one of these approaches, namely the classic double-MST heuristic, in order to obtain a 2β-approximate solution. This improvement is effective, since we show that the double-MST heuristic has in general a performance ratio strictly larger than 2β, and we further show that any alternative elaboration of it cannot lead to a performance ratio better than 2β−ε, for any ε>0. Our theoretical results are complemented with an extensive series of experiments, that show the practical appeal of our approach.