Hardness of Packing, Covering and Partitioning Simple Polygons with Unit Squares

TL;DR

The paper establishes NP-hardness of packing $2\times 2$ axis-aligned squares into simple polygons, including orthogonal, orthogonally convex grids with half-integer coordinates, and extends the hardness to covering and partitioning into small polygons. The authors introduce a novel reduction from Planar-3SAT variants that embed logic into a boundary-following polygon without holes by using reference centers and a row/column abstraction where vertices become rows and edges become columns. They develop two main constructions: a relatively straightforward packing reduction for simple polygons and a far more intricate orthogonally convex construction based on Clover-3SAT, including variable components, PUSH/NEVER gadgets, and complex clause gadgets with membrane rows. The results yield NP-hardness for Small-Cover and Small-Partitioning and demonstrate the first partitioning hardness results for simple polygons, highlighting deep connections between SAT encodings and geometric packing, covering, and partitioning problems with practical implications for grid-based design and manufacturing. The techniques pave the way for further hardness results in related geometric packing problems and suggest directions for exploring convex and non-grid variants, reconfiguration, and approximate algorithms.

Abstract

We show that packing axis-aligned unit squares into a simple polygon $P$ is NP-hard, even when $P$ is an orthogonal and orthogonally convex polygon with half-integer coordinates. It has been known since the early 80s that packing unit squares into a polygon with holes is NP-hard~[Fowler, Paterson, Tanimoto, Inf. Process. Lett., 1981], but the version without holes was conjectured to be polynomial-time solvable more than two decades ago~[Baur and Fekete, Algorithmica, 2001]. Our reduction relies on a new way of reducing from \textsc{Planar-3SAT}. Interestingly, our geometric realization of a planar formula is non-planar. Vertices become rows and edges become columns, with crossings being allowed. The planarity ensures that all endpoints of rows and columns are incident to the outer face of the resulting drawing. We can then construct a polygon following the outer face that realizes all the logic of the formula geometrically, without the need of any holes. This new reduction technique proves to be general enough to also show hardness of two natural covering and partitioning problems, even when the input polygon is simple. We say that a polygon $Q$ is \emph{small} if $Q$ is contained in a unit square. We prove that it is NP-hard to find a minimum number of small polygons whose union is $P$ (covering) and to find a minimum number of pairwise interior-disjoint small polygons whose union is $P$ (partitioning), when $P$ is an orthogonal simple polygon with half-integer coordinates. This is the first partitioning problem known to be NP-hard for polygons without holes, with the usual objective of minimizing the number of pieces.

Figures (64)

• Figure 1: The figure shows schematically how to convert an instance of Monotone-Planar-3SAT into a polygon. Left: The instance of Monotone-Planar-3SAT that we start with. Middle: We realize variables and clauses as rows. Right: We replace the clause rows with dark green auxiliary variables connected to red OR gadgets. The boundary of our constructed polygon is sketched in black.
• Figure 2: The color key.
• Figure 3: The ends of the variable component. The plus position is shown left and the minus is shown right. Some squares in the middle may be pushed up.
• Figure 4: Schematic of how to make a dependency between two variable components. When crossing each variable component in between, the width of the pyramid grows by two squares. The push column is shown in orange. Depending on the position of the bottom variable component $x$, this column may be pushed up (left). Depending on the position of the top variable component $y$, it may be pushed down (right). Since the column can't be pushed both up and down, this creates a constraint between $x$ and $y$.
• Figure 5: A static row. Some squares may be pushed up, but all push left.

Theorems & Definitions (72)

• Theorem 1
• Theorem 2
• Theorem 3
• Lemma 4
• Lemma 5
• proof
• Lemma 6
• proof
• Lemma 7
• proof