Fast Approximations for Metric-TSP via Linear Programming

Abstract: We develop faster approximation algorithms for Metric-TSP building on recent, nearly linear time approximation schemes for the LP relaxation [Chekuri and Quanrud, 2017]. We show that the LP solution can be sparsified via cut-sparsification techniques such as those of Benczur and Karger [2015]. Given a weighted graph $G$ with $m$ edges and $n$ vertices, and $\epsilon &gt; 0$, our randomized algorithm outputs with high probability a $(1+\epsilon)$-approximate solution to the LP relaxation whose support has $\operatorname{O}(n \log n /\epsilon<sup>2)$ edges. The running time of the algorithm is $\operatorname{~O}(m/\epsilon<sup>2)$. This can be generically used to speed up algorithms that rely on the LP. For Metric-TSP, we obtain the following concrete result. For a weighted graph $G$ with $m$ edges and $n$ vertices, and $\epsilon &gt; 0$, we describe an algorithm that outputs with high probability a tour of $G$ with cost at most $(1 + \epsilon) \frac{3}{2}$ times the minimum cost tour of $G$ in time $$\operatorname{~O}(m/\epsilon² + n<sup>{1.5}/\epsilon<sup>3)$$. Previous implementations of Christofides' algorithm [Christofides, 1976] require, for a $\frac{3}{2}$-optimal tour, $\operatorname{~O}(n<sup>{2.5})$ time when the metric is explicitly given, or $$\operatorname{~O}(\min{m^{1.5}, mn+n<sup>{2.5}})$$ time when the metric is given implicitly as the shortest path metric of a weighted graph.