Solving the Minimum Convex Partition of Point Sets with Integer Programming

The partition of a problem into smaller sub-problems satisfying certain properties is often a key ingredient in the design of divide-and-conquer algorithms. For questions related to location, the partition problem can be modeled, in geometric terms, as finding a subdivision of a planar map -- which represents, say, a geographical area -- into regions subject to certain conditions while optimizing some objective function. In this paper, we investigate one of these geometric problems known as the Minimum Convex Partition Problem (MCPP). A convex partition of a point set $P$ in the plane is a subdivision of the convex hull of $P$ whose edges are segments with both endpoints in $P$ and such that all internal faces are empty convex polygons. The MCPP is an NP-hard problem where one seeks to find a convex partition with the least number of faces. We present a novel polygon-based integer programming formulation for the MCPP, which leads to better dual bounds than the previously known edge-based model. Moreover, we introduce a primal heuristic, a branching rule and a pricing algorithm. The combination of these techniques leads to the ability to solve instances with twice as many points as previously possible while constrained to identical computational resources. A comprehensive experimental study is presented to show the impact of our design choices.Comment: 28 pages, 14 figures, submitted for publicatio

The number of edges of many faces in a line segment arrangement

We show that the maximum number of edges bounding m faces in an arrangement of n line segments in the plane is O(m2/3n2/3 +nα(n)+n log m). This improves a previous upper bound of Edelsbrunner et al. [EGS] and almost matches the best known lower bound which is Ω(m2/3n2/3 + nα(n)). In addition, we show that the number of edges bounding any m faces in an arrangement of n line segments with a total of t intersecting pairs is O(m 2/3 t 1/3 + nα ( t n Ω(m 2/3 t 1/3 + nα ( t n}), almost matching the lower bound of)) demonstrated in this paper.)+nmin{log m,log t