Maximum Area Axis-Aligned Square Packings

Given a point set S={s_1,...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 ith 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))>= 1/2 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 NP-complete. This is the first hardness proof for a geometric packing problem where the size of geometric objects in the packing is unrestricted

Anchored Rectangle and Square Packings

For points p_1,...,p_n in the unit square [0,1]^2, an anchored rectangle packing consists of interior-disjoint axis-aligned empty rectangles r_1,...,r_n in [0,1]^2 such that point p_i is a corner of the rectangle r_i (that is, r_i is anchored at p_i) for i=1,...,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, there are point sets for which the area of every anchored rectangle packing is at most 2/3. The maximum area of an anchored square packing is always at least 5/32 and sometimes at most 7/27. The above constructive lower bounds immediately yield constant-factor approximations, of 7/12 -epsilon for rectangles and 5/32 for squares, for computing anchored packings of maximum area in O(n log n) time. We prove that a simple greedy strategy achieves a 9/47-approximation for anchored square packings, and 1/3 for lower-left anchored square packings. Reductions to maximum weight independent set (MWIS) yield a QPTAS and a PTAS for anchored rectangle and square packings in n^{O(1/epsilon)} and exp(poly(log (n/epsilon))) time, respectively

Optimal Packings of Superballs

Dense hard-particle packings are intimately related to the structure of low-temperature phases of matter and are useful models of heterogeneous materials and granular media. Most studies of the densest packings in three dimensions have considered spherical shapes, and it is only more recently that nonspherical shapes (e.g., ellipsoids) have been investigated. Superballs (whose shapes are defined by |x1|^2p + |x2|^2p + |x3|^2p <= 1) provide a versatile family of convex particles (p >= 0.5) with both cubic- and octahedral-like shapes as well as concave particles (0 < p < 0.5) with octahedral-like shapes. In this paper, we provide analytical constructions for the densest known superball packings for all convex and concave cases. The candidate maximally dense packings are certain families of Bravais lattice packings. The maximal packing density as a function of p is nonanalytic at the sphere-point (p = 1) and increases dramatically as p moves away from unity. The packing characteristics determined by the broken rotational symmetry of superballs are similar to but richer than their two-dimensional "superdisk" counterparts, and are distinctly different from that of ellipsoid packings. Our candidate optimal superball packings provide a starting point to quantify the equilibrium phase behavior of superball systems, which should deepen our understanding of the statistical thermodynamics of nonspherical-particle systems.Comment: 28 pages, 16 figure

Compaction of Rods: Relaxation and Ordering in Vibrated, Anisotropic Granular Material

We report on experiments to measure the temporal and spatial evolution of packing arrangements of anisotropic, cylindrical granular material, using high-resolution capacitive monitoring. In these experiments, the particle configurations start from an initially disordered, low-packing-fraction state and under vertical vibrations evolve to a dense, highly ordered, nematic state in which the long particle axes align with the vertical tube walls. We find that the orientational ordering process is reflected in a characteristic, steep rise in the local packing fraction. At any given height inside the packing, the ordering is initiated at the container walls and proceeds inward. We explore the evolution of the local as well as the height-averaged packing fraction as a function of vibration parameters and compare our results to relaxation experiments conducted on spherically shaped granular materials.Comment: 9 pages incl. 7 figure

Periodic Planar Disk Packings

Several conditions are given when a packing of equal disks in a torus is locally maximally dense, where the torus is defined as the quotient of the plane by a two-dimensional lattice. Conjectures are presented that claim that the density of any strictly jammed packings, whose graph does not consist of all triangles and the torus lattice is the standard triangular lattice, is at most $\frac{n}{n+1}\frac{\pi}{\sqrt{12}}$, where $n$ is the number of packing disks. Several classes of collectively jammed packings are presented where the conjecture holds.Comment: 26 pages, 13 figure

Approximating Geometric Knapsack via L-packings

We study the two-dimensional geometric knapsack problem (2DK) in which we are given a set of n axis-aligned rectangular items, each one with an associated profit, and an axis-aligned square knapsack. The goal is to find a (non-overlapping) packing of a maximum profit subset of items inside the knapsack (without rotating items). The best-known polynomial-time approximation factor for this problem (even just in the cardinality case) is (2 + \epsilon) [Jansen and Zhang, SODA 2004]. In this paper, we break the 2 approximation barrier, achieving a polynomial-time (17/9 + \epsilon) < 1.89 approximation, which improves to (558/325 + \epsilon) < 1.72 in the cardinality case. Essentially all prior work on 2DK approximation packs items inside a constant number of rectangular containers, where items inside each container are packed using a simple greedy strategy. We deviate for the first time from this setting: we show that there exists a large profit solution where items are packed inside a constant number of containers plus one L-shaped region at the boundary of the knapsack which contains items that are high and narrow and items that are wide and thin. As a second major and the main algorithmic contribution of this paper, we present a PTAS for this case. We believe that this will turn out to be useful in future work in geometric packing problems. We also consider the variant of the problem with rotations (2DKR), where items can be rotated by 90 degrees. Also, in this case, the best-known polynomial-time approximation factor (even for the cardinality case) is (2 + \epsilon) [Jansen and Zhang, SODA 2004]. Exploiting part of the machinery developed for 2DK plus a few additional ideas, we obtain a polynomial-time (3/2 + \epsilon)-approximation for 2DKR, which improves to (4/3 + \epsilon) in the cardinality case.Comment: 64pages, full version of FOCS 2017 pape

Dense packing on uniform lattices

We study the Hard Core Model on the graphs ${\rm {\bf \scriptstyle G}}$ obtained from Archimedean tilings i.e. configurations in $\scriptstyle \{0,1\}^{{\rm {\bf G}}}$ with the nearest neighbor 1's forbidden. Our particular aim in choosing these graphs is to obtain insight to the geometry of the densest packings in a uniform discrete set-up. We establish density bounds, optimal configurations reaching them in all cases, and introduce a probabilistic cellular automaton that generates the legal configurations. Its rule involves a parameter which can be naturally characterized as packing pressure. It can have a critical value but from packing point of view just as interesting are the noncritical cases. These phenomena are related to the exponential size of the set of densest packings and more specifically whether these packings are maximally symmetric, simple laminated or essentially random packings.Comment: 18 page