Nearly Linear Time Approximation Schemes for Euclidean TSP and other Geometric Problems (1997)
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BibTeX
@MISC{Arora97nearlylinear,
author = {Sanjeev Arora},
title = {Nearly Linear Time Approximation Schemes for Euclidean TSP and other Geometric Problems},
year = {1997}
}
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Abstract
We present a randomized polynomial time approximation scheme for Euclidean TSP in ! 2 that is substantially more efficient than our earlier scheme in [2] (and the scheme of Mitchell [21]). For any fixed c ? 1 and any set of n nodes in the plane, the new scheme finds a (1+ 1 c )-approximation to the optimum traveling salesman tour in O(n(log n) O(c) ) time. (Our earlier scheme ran in n O(c) time.) For points in ! d the algorithm runs in O(n(log n) (O( p dc)) d\Gamma1 ) time. This time is polynomial (actually nearly linear) for every fixed c; d. Designing such a polynomial-time algorithm was an open problem (our earlier algorithm in [2] ran in superpolynomial time for d 3). The algorithm generalizes to the same set of Euclidean problems handled by the previous algorithm, including Steiner Tree, k-TSP, k-MST, etc, although for k-TSP and k-MST the running time gets multiplied by k. We also use our ideas to design nearly-linear time approximation schemes for Euclidean vers...
