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Approximation schemes for NPhard geometric optimization problems: A survey

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


Source: Arora, Sanjeev - Department of Computer Science, Princeton University
Sharir, Micha - School of Computer Science, Tel Aviv University


Collections: Computer Technologies and Information Sciences