 
Summary: Approximation schemes for NPhard
geometric optimization problems: A survey
Sanjeev Arora #
Princeton University
NPhard geometric optimization problems arise in many disciplines. Per
haps the most famous one is the traveling salesman problem (TSP): given n
nodes in # 2 (more generally, in # d ), find the minimum length path that vis
its each node exactly once. If distance is computed using the Euclidean norm
(distance between nodes (x 1 , y 1 ) and (x 2 , y 2 ) is ((x 1 x 2 ) 2
+(y 1 y 2 ) 2 ) 1/2 )
then the problem is called Euclidean TSP. More generally the distance could
be defined using other norms, such as #p norms for any p > 1. All these
are subcases of the more general notion of a geometric norm or Minkowski
norm. We will refer to the version of the problem with a general geometric
norm as geometric TSP.
Some other NPhard geometric optimization problems are Minimum Steiner
Tree (``Given n points, find the smallest network connecting them,''), k
TSP(``Given n points and a number k, find the shortest salesman tour that
visits k points''), kMST (``Given n points and a number k, find the short
est tree that contains k points''), vehicle routing, degree restricted minimum
