Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems

We present a polynomial time approximation scheme for Euclidean TSP in fixed dimensions. For every fixed c > 1 and given any n nodes in ℛ 2 , a randomized version of the scheme finds a (1 + 1/ c )-approximation to the optimum traveling salesman tour in O(n (log n ) O(c) ) time. When the nodes are in ℛ d , the running time increases to O(n (log n ) (O(√ c)) d-1 ). For every fixed c, d the running time is n · poly(log n ), that is nearly linear in n . The algorithmm can be derandomized, but this increases the running time by a factor O(n d ). The previous best approximation algorithm for the problem (due to Christofides) achieves a 3/2-aproximation in polynomial time. We also give similar approximation schemes for some other NP-hard Euclidean problems: Minimum Steiner Tree, k -TSP, and k -MST. (The running times of the algorithm for k -TSP and k -MST involve an additional multiplicative factor k .) The previous best approximation algorithms for all these problems achieved a constant-factor approximation. We also give efficient approximation schemes for Euclidean Min-Cost Matching, a problem that can be solved exactly in polynomial time. All our algorithms also work, with almost no modification, when distance is measured using any geometric norm (such as ℓ p for p ≥ 1 or other Minkowski norms). They also have simple parallel (i.e., NC) implementations.