Approximation Ineffectiveness of a Tour-Untangling Heuristic

TL;DR

This paper analyzes a tour-uncrossing heuristic for the TSP, revealing its worst-case and average-case approximation ratios, and shows that its practical performance exceeds theoretical bounds, highlighting limitations in common analytical methods.

Contribution

The paper provides the first theoretical bounds on the approximation ratios of a tour-uncrossing heuristic for the TSP and evaluates its empirical performance.

Findings

Worst-case approximation ratio is Ω(n)

Average-case approximation ratio is Ω(√n)

Heuristic performs better in practice than theoretical bounds suggest

Abstract

We analyze a tour-uncrossing heuristic for the Travelling Salesperson Problem, showing that its worst-case approximation ratio is $\Omega(n)$ and its average-case approximation ratio is $\Omega(\sqrt{n})$ in expectation. We furthermore evaluate the approximation performance of this heuristic numerically on average-case instances, and find that it performs far better than the average-case lower bound suggests. This indicates a shortcoming in the approach we use for our analysis, which is a rather common approach in the analysis of local search heuristics.