The Traveling Salesman Problem: Low-Dimensionality Implies a Polynomial Time Approximation Scheme

Yair Bartal; Lee-Ad Gottlieb; Robert Krauthgamer

doi:10.1137/130913328

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arXiv · 1112.0699

The Traveling Salesman Problem: Low-Dimensionality Implies a Polynomial Time Approximation Scheme

Yair Bartal·Lee-Ad Gottlieb·Robert Krauthgamer

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Abstract

The Traveling Salesman Problem (TSP) is among the most famous NP-hard optimization problems. We design for this problem a randomized polynomial-time algorithm that computes a (1+eps)-approximation to the optimal tour, for any fixed eps>0, in TSP instances that form an arbitrary metric space with bounded intrinsic dimension. The celebrated results of Arora (A-98) and Mitchell (M-99) prove that the above result holds in the special case of TSP in a fixed-dimensional Euclidean space. Thus, our algorithm demonstrates that the algorithmic tractability of metric TSP depends on the dimensionality of the space and not on its specific geometry. This result resolves a problem that has been open since the quasi-polynomial time algorithm of Talwar (T-04).

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Yair Bartal, Lee-Ad Gottlieb, Robert Krauthgamer. 2011-12-03. The Traveling Salesman Problem: Low-Dimensionality Implies a Polynomial Time Approximation Scheme. https://doi.org/10.1137/130913328

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