Minimum-weight double-tree shortcutting for Metric TSP: Bounding the approximation ratio

TL;DR

This paper investigates the worst-case approximation ratios of the minimum-weight double-tree shortcutting method for the Metric TSP, providing lower bounds in various metric spaces and conjecturing tight bounds for planar Euclidean TSP.

Contribution

It establishes lower bounds on the approximation ratio of the minimum-weight double-tree shortcutting method in different metric spaces and conjectures tight bounds for planar Euclidean TSP.

Findings

Lower bound of 2 in discrete shortest path metric, tight bound.

Lower bound of 1.622 in planar Euclidean metric.

Lower bound of 1.666 in planar Minkowski metric.

Abstract

The Metric Traveling Salesman Problem (TSP) is a classical NP-hard optimization problem. The double-tree shortcutting method for Metric TSP yields an exponentially-sized space of TSP tours, each of which approximates the optimal solution within at most a factor of 2. We consider the problem of finding among these tours the one that gives the closest approximation, i.e.\ the \emph{minimum-weight double-tree shortcutting}. Previously, we gave an efficient algorithm for this problem, and carried out its experimental analysis. In this paper, we address the related question of the worst-case approximation ratio for the minimum-weight double-tree shortcutting method. In particular, we give lower bounds on the approximation ratio in some specific metric spaces: the ratio of 2 in the discrete shortest path metric, 1.622 in the planar Euclidean metric, and 1.666 in the planar Minkowski metric. The first of these lower bounds is tight; we conjecture that the other two bounds are also tight, and in particular that the minimum-weight double-tree method provides a 1.622-approximation for planar Euclidean TSP.