 
Summary: Polynomial Time Approximation Schemes for
Euclidean Traveling Salesman and other Geometric
Problems
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)) d1
). 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
