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 <italic>c</italic> > 1 and given any <italic>n</italic> nodes in <inline-equation><f><sc>R</sc></f> </inline-equation><supscrpt>2</supscrpt>, a randomized version of the scheme finds a (1 + 1/<italic>c</italic>)-approximation to the optimum traveling salesman tour in <italic>O(n</italic>(log <italic>n</italic>)<supscrpt><italic>O(c)</italic>)</supscrpt> time. When the nodes are in <inline-equation><f><sc>R</sc></f> </inline-equation><italic>d</italic>, the running time increases to <italic>O(n</italic>(log <italic>n</italic>)<supscrpt>(O(<inline-equation><f><rad><rcd>d</rcd></rad></f> </inline-equation></supscrpt>c))<supscrpt>d-1</supscrpt>). For every fixed <italic>c, d</italic> the running time is <italic>n</italic> • poly(log<italic>n</italic>), that is <italic>nearly linear</italic> in <italic>n</italic>. The algorithmm can be derandomized, but this increases the running time by a factor <italic>O(n<supscrpt>d</supscrpt></italic>). 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, <italic>k</italic>-TSP, and <italic>k</italic>-MST. (The running times of the algorithm for <italic>k</italic>-TSP and <italic>k</italic>-MST involve an additional multiplicative factor <italic>k</italic>.) 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 <inline-equation><f>ℓ</f> </inline-equation><subscrpt><italic>p</italic></subscrpt> for <italic>p</italic> ≥ 1 or other Minkowski norms). They also have simple parallel (i.e., NC) implementations.

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