Polynomial Time Approximation Schemes for Euclidean TSP and Other Geometric Problems

We present a polynomial time approximation scheme for Euclidean TSP in <2. Given any n nodes in the plane and > 0, the scheme finds a (1 + )-approximation to the optimum traveling salesman tour in time nO(1= ). When the nodes are in <d, the running time increases to nÕ(logd 2 n)= d 1 . The previous best approximation algorithm for the problem (due to Christofides) achieves a 3=2approximation in polynomial time. We also give similar approximation schemes for a host of other Euclidean problems, including Steiner Tree, k-TSP, Minimumdegree-k spanning tree, k-MST, etc. (This list may get longer; our techniques are fairly general.) The previous best approximation algorithms for all these problems achieved a constant-factor approximation. All our algorithms also work, with almost no modification, when distance is measured using any geometric norm (such as `p for p 1 or other Minkowski norms).