 
Summary: Nearly Linear Time Approximation Schemes for Euclidean TSP and
other Geometric Problems
Sanjeev Arora \Lambda
Princeton University
Abstract
We present a randomized polynomial time approx
imation scheme for Euclidean TSP in ! 2 that is sub
stantially 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 algo
rithm 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 polynomialtime algorithm was
