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>0, the scheme finds a (1+ε)-approximation to the optimum traveling salesman tour in time n^0(1/ε). When the nodes are in ℜ^d, the running time increases to n(O˜(log^d-2n)/ε^d-1) The previous best approximation algorithm for the problem (due to Christofides (1976)) 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)