## Polynomial time approximation schemes for Euclidean TSP and other geometric problems (1996)

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Venue: | In Proceedings of the 37th IEEE Symposium on Foundations of Computer Science (FOCS’96 |

Citations: | 317 - 3 self |

### BibTeX

@INPROCEEDINGS{Arora96polynomialtime,

author = {Sanjeev Arora},

title = {Polynomial time approximation schemes for Euclidean TSP and other geometric problems},

booktitle = {In Proceedings of the 37th IEEE Symposium on Foundations of Computer Science (FOCS’96},

year = {1996},

pages = {2--11}

}

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### Abstract

Abstract. We present a polynomial time approximation scheme for Euclidean TSP in fixed dimensions. For every fixed c � 1 and given any n nodes in � 2, a randomized version of the scheme finds a (1 � 1/c)-approximation to the optimum traveling salesman tour in O(n(log n) O(c) ) time. When the nodes are in � d, the running time increases to O(n(log n) (O(�dc))d�1). For every fixed c, d the running time is n � poly(log n), that is nearly linear in n. The algorithm can be derandomized, but this increases the running time by a factor O(n d). 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 some other NP-hard Euclidean problems: Minimum Steiner Tree, k-TSP, and k-MST. (The running times of the algorithm for k-TSP and k-MST involve an additional multiplicative factor k.) 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 �p for p � 1 or other Minkowski norms). They also have simple parallel (i.e., NC) implementations.