Polynomial time approximation schemes for Euclidean TSP and other geometric problems

We present a polynomial time approximation scheme for Euclidean TSP in R². Given any n nodes in the plane and ε>O, the scheme finds a (1+ε)-approximation to the optimum traveling salesman tour in time n^O(1/ε). When the nodes are in R^d, the running time increases to n^{O(log(d-2)n)/ε(d-1)}. 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 a host of other Euclidean problems, including Steiner Tree, k-TSP, Minimum degree-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 l_p for p≥1 or other Minkowski norms).