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

The traveling salesman problem (TSP) is among the most famous NP-hard optimization problems. We design for this problem an algorithm that for any fixed ε>0 computes in randomized polynomial time a (1+ε)-approximation to the optimal tour in TSP instances that form an arbitrary metric space with bounded intrinsic dimension. The celebrated results of Arora [J. ACM, 45 (1998), pp. 753--782] and Mitchell [SIAM J. Comput., 28 (1999), pp. 1298--1309] 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 [Proceedings of the 36th Annual ACM Symposium on Theory of Computing, 2004, pp. 281--290].