Reconfiguration of Polygonal Subdivisions via Recombination

TL;DR

This paper studies how to reconfigure connected polygonal subdivisions, called district maps, through recombination moves that merge and split districts while preserving area, providing bounds on the number of moves needed.

Contribution

It introduces a method to reconfigure one district map into another using a bounded number of recombination moves, with proven tight bounds for small numbers of districts.

Findings

Reconfiguration can be achieved in polylogarithmic moves relative to polygon complexity.

The number of moves needed is tight for small numbers of districts, with lower bounds matching upper bounds.

The approach is constructive and applicable to redistricting problems.

Abstract

Motivated by the problem of redistricting, we study area-preserving reconfigurations of connected subdivisions of a simple polygon. A connected subdivision of a polygon $\mathcal{R}$, called a district map, is a set of interior disjoint connected polygons called districts whose union equals $\mathcal{R}$. We consider the recombination as the reconfiguration move which takes a subdivision and produces another by merging two adjacent districts, and by splitting them into two connected polygons of the same area as the original districts. The complexity of a map is the number of vertices in the boundaries of its districts. Given two maps with $k$ districts, with complexity $O(n)$, and a perfect matching between districts of the same area in the two maps, we show constructively that $(\log n)^{O(\log k)}$ recombination moves are sufficient to reconfigure one into the other. We also show that $\Omega(\log n)$ recombination moves are sometimes necessary even when $k=3$, thus providing a tight bound when $k=O(1)$.