Covering and Partitioning Complex Objects with Small Pieces

TL;DR

This paper introduces a PTAS for covering and partitioning polygons with small axis-aligned squares, improving previous approximation algorithms, and proves NP-hardness for the 3D version of the problem.

Contribution

It establishes the first PTAS for 2D polygon covering and partitioning with small squares and proves NP-hardness of approximation in 3D.

Findings

A local search algorithm yields a 1+O(1/√k) approximation for large k.

The problems of covering and partitioning are shown to be equivalent.

NP-hardness of approximating the problem in 3D polyhedra.

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.