Fast minimum-weight double-tree shortcutting for Metric TSP: Is the best one good enough?

TL;DR

This paper introduces an improved algorithm for the minimum-weight double-tree shortcutting in Metric TSP, enabling larger instance solving and demonstrating its effectiveness as a high-quality heuristic.

Contribution

An improved algorithm with better time and memory complexity for finding the optimal double-tree shortcutting in Metric TSP, especially for instances with small maximum node degree.

Findings

The new algorithm is faster and more memory-efficient for small node degrees.

It enables solving larger TSP instances than previous methods.

The method offers a competitive tradeoff between solution quality and computational time.

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}. Burkard et al. gave an algorithm for this problem, running in time $O(n^3+2^d n^2)$ and memory $O(2^d n^2)$, where $d$ is the maximum node degree in the rooted minimum spanning tree. We give an improved algorithm for the case of small $d$ (including planar Euclidean TSP, where $d \leq 4$), running in time $O(4^d n^2)$ and memory $O(4^d n)$. This improvement allows one to solve the problem on much larger instances than previously attempted. Our computational experiments suggest that in terms of the time-quality tradeoff, the minimum-weight double-tree shortcutting method provides one of the best known tour-constructing heuristics.