Partitioning a Polygon Into Small Pieces

TL;DR

It seems out of reach to compute optimal partitions for simple polygons; for most of them, even in extremely restricted cases such as when $P$ is a square, the O(1)-approximation algorithms for these problems are developed, which means that the number of pieces in the produced partition is at most a constant factor larger than the cardinality of a minimum partition.

Abstract

We study the problem of partitioning a given simple polygon $P$ into a minimum number of connected polygonal pieces, each of bounded size. We describe a general technique for constructing such partitions that works for several notions of `bounded size,' namely that each piece must be contained in an axis-aligned or arbitrarily rotated unit square or a unit disk, or that each piece has bounded perimeter, straight-line diameter or geodesic diameter. The problems are motivated by practical settings in manufacturing, finite element analysis, collision detection, vehicle routing, shipping and laser capture microdissection. The version where each piece should be contained in an axis-aligned unit square is already known to be NP-hard [Abrahamsen and Stade, FOCS, 2024], and the other versions seem no easier. Our main result is to develop constant-factor approximation algorithms, which means that the number of pieces in the produced partition is at most a constant factor larger than the cardinality of an optimal partition. Existing algorithms [Damian and Pemmaraju, Algorithmica, 2004] do not allow Steiner points, which means that all corners of the produced pieces must also be corners of $P$. This has the disappointing consequence that a partition often does not exist, whereas our algorithms always produce meaningful partitions. Furthermore, an optimal partition without Steiner points may require $Ω(n)$ pieces for polygons with $n$ corners where a partition consisting of just $2$ pieces exists when Steiner points are allowed. Other existing algorithms [Arkin, Das, Gao, Goswami, Mitchell, Polishchuk and Tóth, ESA, 2020] only allow $P$ to be split along chords (and aim to minimize the number of chords instead of the number of pieces), whereas we make no constraints on the boundaries of the pieces.

Figures (22)

• Figure 1: The spirals indicate that we cannot use an algorithm for aligned square partitions or straight diameter partitions to get an algorithm for the other problem.
• Figure 2: Two disk partitions of the same polygon $P$. The corners of $P$ are contained in two concentric circular arcs. The concave arc has radius $1+\varepsilon$ for a small value $\varepsilon>0$. The convex arc has radius $3$. When Steiner points are not allowed, where $\Omega(n)$ pieces are needed in a disk partition. With Steiner points, $2$ pieces are enough.
• Figure 3: An aligned square partition. The boundary pieces are blue, and they cover all of the boundary, although they are in some places degenerate so that they cannot be seen. The yellow, red and green pieces are so-called complete, edge and chip pieces, respectively. Color codes: 1 blue, 2 yellow, 3 red, 4 green.
• Figure 4: Left: A triangulated polygon. Middle: The triangulation we obtain after adding all medians of the triangles. The Hamiltonian cycle $C$ is also shown. Right: An area partition into five pieces produced by our algorithm.
• Figure 5: The blue pieces form a boundary partition of a polygon $P$. The pieces $Q_1$ and $Q_2$ are degenerate and follow the boundary of $P$ to the point $p$. It is shown how we can make a small perturbation so that the intersection of each piece with $\partial P$ is just a single interval.

Theorems & Definitions (19)

• Theorem 1
• Theorem 2
• Lemma 3
• proof
• Lemma 4
• proof
• Lemma 5
• proof
• Lemma 6
• proof