Covering and Partitioning Complex Objects with Small Pieces

2026-03-24

AI Summary

This work addresses the problem of covering or partitioning a two-dimensional polygon with holes using the minimum number of connected subpolygons, each contained within an axis-aligned unit square, and extends the study to three dimensions. By integrating local search techniques with tools from computational geometry and complexity theory, the paper presents the first polynomial-time approximation scheme (PTAS) for the two-dimensional case with holes, achieving an approximation ratio of \(1 + O(1/\sqrt{k})\), which significantly improves upon the previously known 13-approximation limited to hole-free polygons. The study also establishes the equivalence between covering and partitioning in terms of optimal solution size. In three dimensions, the problem is shown to be NP-hard to approximate within any logarithmic factor, thereby delineating its computational complexity boundary.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Applications of cryptography
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

We study the problems of covering or partitioning a polygon $P$ (possibly with holes) using a minimum number of small pieces, where a small piece is a connected sub-polygon contained in an axis-aligned unit square. For covering, we seek to write $P$ as a union of small pieces, and in partitioning, we furthermore require the pieces to be pairwise interior-disjoint. We show that these problems are in fact equivalent: Optimum covers and partitions have the same number of pieces. For covering, a natural local search algorithm repeatedly attempts to replace $k$ pieces from a candidate cover with $k-1$ pieces. In two dimensions and for sufficiently large $k$, we show that when no such swap is possible, the cover is a $1+O(1/\sqrt k)$-approximation, hence obtaining the first PTAS for the problem. Prior to our work, the only known algorithm was a $13$-approximation that only works for polygons without holes [Abrahamsen and Rasmussen, SODA 2025]. In contrast, in the three dimensional version of the problem, for a polyhedron $P$ of complexity $n$, we show that it is NP-hard to approximate an optimal cover or partition to within a factor that is logarithmic in $n$, even if $P$ is simple, i.e., has genus $0$ and no holes.

Problem

• covering
• partitioning
• polygon
• small pieces
• approximation

Innovation

• polygon covering
• local search
• PTAS
• computational geometry
• inapproximability