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 (log n)(O(c))) time. When the nodes are in R-d, the running time increases to O(n(log n)((O(root dc))d-1)). For every fixed c, d the running time is n poly(log n), that is nearly linear in n. The algorithm can be derandomized, but this increases the running time by a factor O(nd). The previous best approximation algorithm for the problem (due to Christofides) achieves a 3/2-approximation in polynomial time. We also give similar approximation schemes for some other NP-hard Euclidean problems: Minimum Steiner Tree, R-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 greater than or equal to 1 or other Minkowski norms). They also have simple parallel (i.e., NC) implementations.