Given a point set P , are k closed convex polygons sufficient to partition the convex hull of P , such that the interior of a convex set contains no point in P? What if the vertices of these polygons are constrained to be points of P ? We show that the first decision problem is NP-hard, and the second NP-complete } .
We consider the Minimum Convex Partition problem: Given a set P of n points in the plane, draw a plane graph G on P, with positive minimum degree, such that G partitions the convex hull of P into a minimum number of convex faces. We show that Minimum Convex Partition is NP-hard, and we give a O(sqrt n}log n)-approximation algorithm running in quadratic time. The latter result is obtained by relating the problem to another geometric problem: Given a set of points P, cover P with a minimum number of lines. A constant-factor approximation algorithm was shown for Minimum Convex Partition by Knauer and Spillner, for point sets where no three points are on a line (SWAT 2006). This restriction is crucial for the algorithm to work. The complexity of the problem under this restriction remains open .
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Given a point set P , are k closed convex polygons sufficient to partition the convex hull of
We consider the Minimum Convex Partition problem: Given a set
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, such that the interior of a convex set contains no point in P? What if the vertices of these polygons are constrained to be points of P ?
of n points in the plane, draw a plane graph G on P, with positive minimum degree, such that G partitions the convex hull of P into a minimum number of convex faces.
R
the first decision problem
Minimum Convex Partition
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the second NP-complete
we give a O(sqrt
A
n
A
log n)-approximation algorithm running in quadratic time. The latter result is obtained by relating the problem to another geometric problem: Given a set of points P, cover P with a minimum number of lines. A constant-factor approximation algorithm was shown for Minimum Convex Partition by Knauer and Spillner, for point sets where no three points are on a line (SWAT 2006). This restriction is crucial for the algorithm to work. The complexity of the problem under this restriction remains open