Approximating the Metric TSP in Linear Time

Abstract

Given a metric graph G = (V,E) of n vertices, i.e., a complete graph with an edge cost function c:V ×V →ℝ_≥ 0 satisfying the triangle inequality, the metricity degree of G is defined as $\beta=\max_{x,y,z \in V} \big\{ \frac{c(x,y)}{c(x,z)+c(y,z)}\big\} \in \big[\frac{1}{2},1\big]$. 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 ^2.5 log^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 the 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 enhance –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 that 2 β, and we further show that any re-elaboration of the shortcutting phase therein provided, cannot lead to a performance ratio better than 2β.