Bicriteria Polygon Aggregation with Arbitrary Shapes

TL;DR

This paper presents a polynomial-time algorithm for bicriteria polygon aggregation with arbitrary shapes, optimizing for area and perimeter, and introduces approximation methods for variants with polygonal boundaries, validated on real-world data.

Contribution

It provides the first efficient algorithm for unrestricted boundary shapes in polygon aggregation, characterizes optimal solutions, and offers approximation techniques for practical variants.

Findings

Polynomial-time algorithm for arbitrary boundary shapes.

Optimal solutions have boundaries composed of polygon edges and circular arcs.

Approximate solutions are visually similar and near-optimal on real-world data.

Abstract

We study the problem of aggregating polygons by covering them with disjoint representative regions, thereby inducing a clustering of the polygons. Our objective is to minimize a weighted sum of the total area and the total perimeter of the regions. This problem has applications in cartographic map generalization and urban analytics. Here, the polygons represent building footprints and the clusters may represent urban areas. Previous approaches forced the boundaries of the regions to come from a fixed subdivision of the plane, which allows the optimal solution (restricted in this way) to be found from a minimum cut in a dual graph. It is natural to ask whether the problem can still be solved efficiently if this restriction is removed, allowing output regions to be bounded by arbitrary curves. We provide a positive answer in the form of a polynomial-time algorithm. Additionally, we fully characterize the optimal solutions by showing that their boundaries are composed of input polygon edges and circular arcs of constant radius. Since some applications favor straight edges, we also study two problem variants in which the output regions must be polygons, but are not restricted to have boundaries from a fixed subdivision. In the first variant, region vertices must lie on the boundaries of the input polygons. The second variant requires them to be vertices of the input polygons. We show that both variants can be approximated up to a constant factor in polynomial time by altering an optimal solution for the unrestricted problem. Our experimental evaluation on real-world building footprints demonstrates that these approximate solutions are visually similar to the optimal unrestricted ones and achieve near-optimal objective values.