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Approximation Ineffectiveness of a Tour-Untangling Heuristic
Bodo Manthey, Jesse van Rhijn · 2023-02-22 · via math.PR updates on arXiv.org

We analyze a tour-uncrossing heuristic for the Travelling Salesperson Problem, showing that its worst-case approximation ratio is $Ω(n)$ and its average-case approximation ratio is $Ω(\sqrt{n})$ 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.

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