## Approximation schemes for NP-hard geometric optimization problems: A survey (2003)

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Venue: | Mathematical Programming |

Citations: | 38 - 2 self |

### BibTeX

@ARTICLE{Arora03approximationschemes,

author = {Sanjeev Arora},

title = {Approximation schemes for NP-hard geometric optimization problems: A survey},

journal = {Mathematical Programming},

year = {2003},

volume = {97},

pages = {2003}

}

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### Abstract

NP-hard geometric optimization problems arise in many disciplines. Perhaps the most famous one is the traveling salesman problem (TSP): given n nodes in ℜ 2 (more generally, in ℜ d), find the minimum length path that visits each node exactly once. If distance is computed using the Euclidean norm (distance between nodes (x1, y1) and (x2, y2) is ((x1−x2) 2 +(y1−y2) 2) 1/2) then the problem is called Euclidean TSP. More generally the distance could be defined using other norms, such as ℓp 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 NP-hard 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”), k-MST (“Given n points and a number k, find the shortest tree that contains k points”), vehicle routing, degree restricted minimum