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arXiv · 2310.03222

The bright side of simple heuristics for the TSP

Alan Frieze·Wesley Pegden
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Abstract

The greedy and nearest-neighbor TSP heuristics can both have $\log n$ approximation factors from optimal in worst case, even just for $n$ points in Euclidean space. In this note, we show that this approximation factor is only realized when the optimal tour is unusually short. In particular, for points from any fixed $d$-Ahlfor's regular metric space (which includes any $d$-manifold like the $d$-cube $[0,1]^d$ in the case $d$ is an integer but also fractals of dimension $d$ when $d$ is real-valued), our results imply that the greedy and nearest-neighbor heuristics have \emph{additive} errors from optimal on the order of the \emph{optimal} tour length through \emph{random} points in the same space, for $d>1$.

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Alan Frieze, Wesley Pegden. 2023-10-05. The bright side of simple heuristics for the TSP. https://arxiv.org/abs/2310.03222

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