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.

将问题划分为满足某些属性的更小的子问题通常是分治算法设计中的一个关键因素。对于与位置相关的问题，分区问题可以用几何术语来建模，即在优化某些目标函数的同时，将平面地图（例如，一个地理区域）细分为受某些条件约束的区域。在本文中，我们研究了这些几何问题之一，称为最小凸分割问题 ( mcpp )。甲凸分区点集的P中的平面的凸包的细分P，其边缘片段，以两个端点P并且所有内部面都是空的凸多边形。所述MCPP是其中一个目的是找到具有面的最少数量的凸分区的NP-hard问题。

我们为mcpp提出了一种新的基于多边形的整数规划公式，它比以前已知的基于边的模型产生更好的双重边界。此外，我们引入了原始启发式、分支规则和定价算法。这些技术的结合使我们能够在受限于相同计算资源的情况下解决具有两倍于以前的点的实例的能力。提出了一项全面的实验研究，以显示我们的设计选择的影响。