Approximation Ineffectiveness of a Tour-Untangling Heuristic

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.