We provide a tight result for a fundamental problem arising from packing squares into a circular container: The critical density of packing squares into a disk is $δ=\frac{8}{5π}\approx 0.509$. This implies that any set of (not necessarily equal) squares of total area $A \leq \frac{8}{5}$ can always be packed into a disk with radius 1; in contrast, for any $\varepsilon>0$ there are sets of squares of total area $\frac{8}{5}+\varepsilon$ that cannot be packed, even if squares may be rotated. This settles the last (and arguably, most elusive) case of packing circular or square objects into a circular or square container: The critical densities for squares in a square $\left(\frac{1}{2}\right)$, circles in a square $\left(\fracπ{(3+2\sqrt{2})}\approx 0.539\right)$ and circles in a circle $\left(\frac{1}{2}\right)$ have already been established, making use of recursive subdivisions of a square container into pieces bounded by straight lines, or the ability to use recursive arguments based on similarity of objects and container; neither of these approaches can be applied when packing squares into a circular container. Our proof uses a careful manual analysis, complemented by a computer-assisted part that is based on interval arithmetic. Beyond the basic mathematical importance, our result is also useful as a blackbox lemma for the analysis of recursive packing algorithms. At the same time, our approach showcases the power of a general framework for computer-assisted proofs, based on interval arithmetic.
We provide a tight result for a fundamental problem arising from packing disks into a circular container: The critical density of packing disks in a disk is 0.5. This implies that any set of (not necessarily equal) disks of total area $δ\leq 1/2$ can always be packed into a disk of area 1; on the other hand, for any $\varepsilon>0$ there are sets of disks of area $1/2+\varepsilon$ that cannot be packed. The proof uses a careful manual analysis, complemented by a minor automatic part that is based on interval arithmetic. Beyond the basic mathematical importance, our result is also useful as a b…
In the classic circle packing problem, one asks whether a given set of circles can be packed into a given container. Packing problems like this have been shown to be $\mathsf{NP}$-hard. In this paper, we present new sufficient conditions for packing circles into square and triangular containers, using only the sum of the circles' areas: For square containers, it is possible to pack any set of circles with a combined area of up to approximately 53.90% of the square's area. And when the container is a right or obtuse triangle, any set of circles whose combined area does not exceed the triangle's…
In this paper, we show an improved bound and new algorithm for the online square-into-square packing problem. This two-dimensional packing problem involves packing an online sequence of squares into a unit square container without any two squares overlapping. The goal is to find the largest area $α$ such that any set of squares with total area $α$ can be packed. We show an algorithm that can pack any set of squares with total area $α\leq 3/8$ into a unit square in an online setting, improving the previous bound of $11/32$.
Consider a set $P$ of $n$ points on the boundary of an axis-aligned square $Q$. We study the boundary-anchored packing problem on $P$ in which the goal is to find a set of interior-disjoint axis-aligned rectangles in $Q$ such that each rectangle is anchored (has a corner at some point in $P$), each point in $P$ is used to anchor at most one rectangle, and the total area of the rectangles is maximized. Here, a rectangle is anchored at a point $p$ in $P$ if one of its corners coincides with $p$. In this paper, we show how to solve this problem in time linear in $n$, provided that the points of $…
The problem widely known as Moser's Square Packing Problem asks for the smallest area $A$ such that for any set $S$ of squares of total area $1$, there exists a rectangle $R$ of area $A$ into which the squares in $S$ permit an internally-disjoint, axis-parallel packing. It was formulated by Moser in 1966 and remains unsolved so far. The best known lower bound of $\frac{2+\sqrt{3}}{3}\leq A$ is due to Novotný and has been shown to be sufficient for up to $11$ squares by Platz, while Hougardy and Ilhan have established that $A < 1.37$. In this paper, we reduce Moser's Square Packing Problem to a…
Given a point set $S=\{s_1,\ldots , s_n\}$ in the unit square $U=[0,1]^2$, an anchored square packing is a set of $n$ interior-disjoint empty squares in $U$ such that $s_i$ is a corner of the $i$th square. The reach $R(S)$ of $S$ is the set of points that may be covered by such a packing, that is, the union of all empty squares anchored at points in $S$. It is shown that area$(R(S))\geq \frac12$ for every finite set $S\subset U$, and this bound is the best possible. The region $R(S)$ can be computed in $O(n\log n)$ time. Finally, we prove that finding a maximum area anchored square packing is …
For points $p_1,\ldots , p_n$ in the unit square $[0,1]^2$, an \emph{anchored rectangle packing} consists of interior-disjoint axis-aligned empty rectangles $r_1,\ldots , r_n\subseteq [0,1]^2$ such that point $p_i$ is a corner of the rectangle $r_i$ (that is, $r_i$ is \emph{anchored} at $p_i$) for $i=1,\ldots, n$. We show that for every set of $n$ points in $[0,1]^2$, there is an anchored rectangle packing of area at least $7/12-O(1/n)$, and for every $n\in \mathbf{N}$, there are point sets for which the area of every anchored rectangle packing is at most $2/3$. The maximum area of an anchored…
We analyze the problem of packing squares in an online fashion: Given a semi-infinite strip of width 1 and an unknown sequence of squares of side length in [0,1] that arrive from above, one at a time. The objective is to pack these items as they arrive, minimizing the resulting height. Just like in the classical game of Tetris, each square must be moved along a collision-free path to its final destination. In addition, we account for gravity in both motion (squares must never move up) and position (any final destination must be supported from below). A similar problem has been considered befor…
In this paper, we consider the following geometric puzzle whose origin was traced to Allan Freedman \cite{croft91,tutte69} in the 1960s by Dumitrescu and T{ó}th \cite{adriancasaba2011}. The puzzle has been popularized of late by Peter Winkler \cite{Winkler2007}. Let $P_{n}$ be a set of $n$ points, including the origin, in the unit square $U = [0,1]^2$. The problem is to construct $n$ axis-parallel and mutually disjoint rectangles inside $U$ such that the bottom-left corner of each rectangle coincides with a point in $P_{n}$ and the total area covered by the rectangles is maximized. We would te…
A polyomino is a polygonal region with axis parallel edges and corners of integral coordinates, which may have holes. In this paper, we consider planar tiling and packing problems with polyomino pieces and a polyomino container $P$. We give two polynomial time algorithms, one for deciding if $P$ can be tiled with $k\times k$ squares for any fixed $k$ which can be part of the input (that is, deciding if $P$ is the union of a set of non-overlapping $k\times k$ squares) and one for packing $P$ with a maximum number of non-overlapping and axis-parallel $2\times 1$ dominos, allowing rotations by $9…
In 1967, Moon and Moser proved a tight bound on the critical density of squares in squares: any set of squares with a total area of at most 1/2 can be packed into a unit square, which is tight. The proof requires full knowledge of the set, as the algorithmic solution consists in sorting the objects by decreasing size, and packing them greedily into shelves. Since then, the online version of the problem has remained open; the best upper bound is still 1/2, while the currently best lower bound is 1/3, due to Han et al. (2008). In this paper, we present a new lower bound of 11/32, based on a dyna…
We consider the online problem of packing circles into a square container. A sequence of circles has to be packed one at a time, without knowledge of the following incoming circles and without moving previously packed circles. We present an algorithm that packs any online sequence of circles with a combined area not larger than 0.350389 0.350389 of the square's area, improving the previous best value of π/10 \approx 0.31416; even in an offline setting, there is an upper bound of π/(3 + 2 \sqrt{2}) \approx 0.5390. If only circles with radii of at least 0.026622 are considered, our algorithm ach…
The Split Packing algorithm \cite{splitpacking_ws, splitpackingsoda, splitpacking} is an offline algorithm that packs a set of circles into triangles and squares up to critical density. In this paper, we develop an online alternative to Split Packing to handle an online sequence of insertions and deletions, where the algorithm is allowed to reallocate circles into new positions at a cost proportional to their areas. The algorithm can be used to pack circles into squares and right angled triangles. If only insertions are considered, our algorithm is also able to pack to critical density, with a…
We address the NP-hard problem of finding a non-overlapping dense packing pattern for n Unequal Circle items in a two-dimensional Square Container (PUC-SC) such that the size of the container is minimized. Based on our previous work on an Action Space based Global Optimization (ASGO) that approximates each circle item as a square item to efficiently find the large unoccupied spaces, we propose an optimization algorithm based on the Partitioned Action Space and Partitioned Circle Items (PAS-PCI). The PAS is to partition the narrow action space on the long side to find two equal action spaces to…
We show that deciding whether a given set of circles can be packed into a rectangle, an equilateral triangle, or a unit square are NP-hard problems, settling the complexity of these natural packing problems. On the positive side, we show that any set of circles of total area 1 can be packed into a square of size 4/\sqrt{pi}=2.2567... These results are motivated by problems arising in the context of origami design.
We study the packing of a large number of congruent and non--overlapping circles inside a regular polygon. We have devised efficient algorithms that allow one to generate configurations of $N$ densely packed circles inside a regular polygon and we have carried out intensive numerical experiments spanning several polygons (the largest number of sides considered here being $16$) and up to $200$ circles ($400$ circles in the special cases of the equilateral triangle and the regular hexagon) . Some of the configurations that we have found possibly are not global maxima of the packing fraction, par…
This work revolves around the two following questions: Given a convex body $C\subset\mathbb{R}^d$, a positive integer $k$ and a finite set $S\subset\mathbb{R}^d$ (or a finite Borel measure $μ$ on $\mathbb{R}^d$), how many homothets of $C$ are required to cover $S$ if no homothet is allowed to cover more than $k$ points of $S$ (or have measure larger than $k$)? How many homothets of $C$ can be packed if each of them must cover at least $k$ points of $S$ (or have measure at least $k$)? We prove that, so long as $S$ is not too degenerate, the answer to both questions is $Θ_d(\frac{|S|}{k})$, wher…
