Summary: Polynomial Time Approximation Schemes for Euclidean TSP and other Geometric
Sanjeev Arora \Lambda
We present a polynomial time approximation scheme for
Euclidean TSP in ! 2 . Given any n nodes in the plane
and ffl ? 0, the scheme finds a (1 + ffl)approximation
to the optimum traveling salesman tour in time n O(1=ffl) .
When the nodes are in ! d , the running time increases to
O(log d\Gamma2 n)=ffl d\Gamma1
. The previous best approximation algo
rithm for the problem (due to Christofides) achieves a 3=2
approximation in polynomial time.
We also give similar approximation schemes for a host
of other Euclidean problems, including Steiner Tree, kTSP,
Minimumdegreek spanning tree, kMST, etc. (This list may
get longer; our techniques are fairly general.) The previ
ous best approximation algorithms for all these problems