Mittagsseminar (in cooperation with A. Steger, D. Steurer and B. Sudakov)

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

Duration: 30 minutes

Location: Zoom: conference room

Speaker: Nicolas Grelier

Hardness and Approximation of Minimum Convex Partition

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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