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Covering and Partitioning Complex Objects with Small Pieces

Anders Aamand, Mikkel Abrahamsen, Reilly Browne, Mayank Goswami, Prahlad Narasimhan Kasthurirangan, Linda Kleist, Joseph S. B. Mitchell, Valentin Polishchuk, Jack Stade

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.

Covering and Partitioning Complex Objects with Small Pieces

Abstract

We study the problems of covering or partitioning a polygon (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 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 pieces from a candidate cover with pieces. In two dimensions and for sufficiently large , we show that when no such swap is possible, the cover is a -approximation, hence obtaining the first PTAS for the problem. Prior to our work, the only known algorithm was a -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 of complexity , we show that it is NP-hard to approximate an optimal cover or partition to within a factor that is logarithmic in , even if is simple, i.e., has genus and no holes.
Paper Structure (14 sections, 21 theorems, 2 equations, 4 figures)

This paper contains 14 sections, 21 theorems, 2 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Related Work
  3. Problem Formulation
  4. A PTAS for Small-Cover in 2 dimensions
  5. Fundamentals for Local Search
  6. Polygons with Small Diameter
  7. Polygons with Large Diameter
  8. Equivalence of Small Covers and Partitions
  9. Hardness for Small-Cover in 3 dimensions
  10. Proof of \ref{['lem:nondegenerate']}
  11. Proof of \ref{['thm:sparse-reduction']}
  12. Constructing a set of simple instances
  13. Approximating the separate instances suffices
  14. Approximating the separate instances

Key Result

Lemma 1

For a universal constant $C$ the following holds: Let $X$ be a (finite) set of points in $\mathbb{R}^2$ and suppose that $\mathcal{Q}=\{Q_1, \dots, Q_q\}$ and $\mathcal{R}=\{R_1, \dots, R_r\}$ are sets of connected regions such that $\mathcal{Q}$ and $\mathcal{R}$ each cover $X$. If the regions in $

Figures (4)

  • Figure 1: A comb-shaped polygon might require an arbitrarily large number of pieces for a small cover, even if it fits in a box of size $1\times (1+\varepsilon)$.
  • Figure 2: Given a finite set of complete small pieces that cover $P$, we obtain an equivalent instance of non-piercing geometric set cover. Left: a polygon and two complete small pieces. Top middle: the polygon and pieces cut the polygon into $4$ cells, and we place a point in each cell. These pieces are piercing, since removing the red piece from the blue piece would disconnect it. Additionally, the red piece has a hole. Bottom middle: modifying the pieces to remove the hole and make them non-piercing. The hole in the red piece is filled, and an interval of the boundary is offset inwards so as to not pierce the blue piece. Right: removing the modified blue piece from the red piece (top) or the modified red piece from the blue piece (bottom). The remaining parts are connected, so the modified pieces are non-piercing.
  • Figure 3: The combinatorial structure of a set of complete small pieces. Left: The connected components of $P^\circ \cap S$ are numbered. Right: The centers of the bounding squares of the two pieces are marked with squares. The intersection of lines $\ell_1$ and $\ell_2$ lies above $\ell_3$.
  • Figure 13: A corridor $C_i$ in gray with two vertical sides. The corridor $C_i$ is further partitioned into sections $S_i^j$ with two vertical sides all having length exactly $1/\varepsilon$.

Theorems & Definitions (21)

  • Lemma 1: NonpiercingPTAS
  • Lemma 2
  • Lemma 3
  • Corollary 4
  • Lemma 5
  • Lemma 6
  • Theorem 7
  • Theorem 8
  • Corollary 9
  • Theorem 10
  • ...and 11 more