Bicriteria Polygon Aggregation with Arbitrary Shapes

2025-07-15

AI Summary

This paper addresses the polygon aggregation problem: clustering an input family of polygons by covering them with disjoint representative regions, minimizing a weighted objective (total area + λ × total perimeter), where representative boundaries may be arbitrary smooth curves—not restricted to fixed planar subdivisions. We present the first polynomial-time optimal algorithm, proving that optimal boundaries consist exclusively of segments from input polygon edges and circular arcs of fixed radius. Our method builds upon a dual-graph minimum-cut framework, integrating geometric structural analysis with approximation techniques to support general curve modeling and multiple polygonal output variants. Experiments on real-world building footprint data demonstrate that our approximate solutions achieve near-optimal visual quality and objective values while maintaining high computational efficiency—making the approach suitable for map generalization and urban spatial analysis.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

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.

Problem

• Aggregating polygons into disjoint regions with minimal area and perimeter
• Solving the problem efficiently without fixed subdivision constraints
• Approximating polygonal variants with constant-factor guarantees in polynomial time

Innovation

• Polynomial-time algorithm for arbitrary region boundaries
• Optimal solutions use polygon edges and circular arcs
• Constant-factor approximation for polygonal output variants