Department of Computer Science | Institute of Theoretical Computer Science | CADMO

Prof. Emo Welzl and Prof. Bernd Gärtner

Mittagsseminar Talk Information |

**Date and Time**: Thursday, May 14, 2020, 12:15 pm

**Duration**: 30 minutes

**Location**: Zoom: conference room

**Speaker**: Nicolas Grelier

We consider the Minimum Convex Partition problem: Given a set P of n points in the plane, draw a plane a 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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