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Polynomial Time Approximation Schemes for Euclidean Traveling Salesman and other Geometric

Summary: Polynomial Time Approximation Schemes for
Euclidean Traveling Salesman and other Geometric
Sanjeev Arora
Princeton University
Association for Computing Machinery, Inc., 1515 Broadway, New York, NY 10036, USA
Tel: (212) 555­1212; Fax: (212) 555­2000
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), i.e., 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 3/2­approximation in polynomial time.
We also give similar approximation schemes for some other NP­hard Euclidean problems: Mini­
mum Steiner Tree, k­TSP, and k­MST. (The running times of the algorithm for k­TSP and k­MST
involve an additional multiplicative factor k.) The previous best approximation algorithms for all
these problems achieved a constant­factor approximation. We also give e#cient approximation
schemes for Euclidean Min­Cost Matching, a problem that can be solved exactly in polynomial


Source: Arora, Sanjeev - Department of Computer Science, Princeton University


Collections: Computer Technologies and Information Sciences