Solving the Minimum Convex Partition of Point Sets with Integer Programming

TL;DR

This paper introduces a new integer programming approach to solve the NP-hard Minimum Convex Partition Problem more efficiently, enabling solutions for larger point sets than previously possible.

Contribution

A novel polygon-based integer programming formulation for MCPP, along with heuristics and algorithms that improve solution size and computational efficiency.

Findings

Better dual bounds than previous models

Able to solve larger instances with same resources

Significant experimental improvements

Abstract

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.