Packing similar triangles into a triangle

There exists a triangle T and a number \frac{1}{2}$$ " align="middle" border="0"> such that any sequence of triangles similar to T with total area not greater than times the area of T can be packed into T.     

Packing two copies of a sparse graph into a graph with restrained maximum degree

We show that if a tree T is not a star, then there is an embedding σ of T in the complement of T such that the maximum degree of T∪σ(T) is at most Δ(T)+2. We also show that if G is a graph of order n with n?1 edges, then with several exceptions, there exists an embedding σ of G in the complement of G such that the maximum degree of G∪σ(G) is at most Δ(G)+3. Both results are sharp in the sense that neither of Δ(T)+2 and Δ(G)+3 can be reduced. From these two results, we deduce two corollaries on packings of three graphs. © 2009 Wiley Periodicals, Inc. J Graph Theory 62: 178–187, 2009     

Packing trees into planar graphs

In this study, we provide methods for drawing a tree with n vertices on a convex polygon, without crossings and using the minimum number of edges of the polygon. We apply the results to obtain planar packings of two trees in some specific cases. © 2002 Wiley Periodicals, Inc. J Graph Theory 40: 172–181, 2002     

Packing Rectangles into the Unit Square

Every sequence of rectangles of side lengths not greater than 1, whose total area is smaller than or equal to can be packed into the unit square.     

Covering moving points with anchored disks

Consider a set of mobile clients represented by n points in the plane moving at constant speed along n different straight lines. We study the problem of covering all mobile clients using a set of k disks centered at k fixed centers. Each disk exists only at one instant and while it does, covers any client within its coverage radius. The task is to select an activation time and a radius for each disk such that every mobile client is covered by at least one disk. In particular, we study the optimization problem of minimizing the maximum coverage radius. First we prove that, although the static version of the problem is polynomial, the kinetic version is NP-hard. Moreover, we show that the problem is not approximable by a constant factor (unless P = NP). We then present a generic framework to solve it for fixed values of k, which in turn allows us to solve more general optimization problems. Our algorithms are efficient for constant values of k.     

Packing two forests into a bipartite graph

For two integers a and b, we say that a bipartite graph G admits an (a, b)-bipartition if G has a bipartition (X, Y) such that |X| = a and |Y| = b. We say that two bipartite graphs G and H are compatible if, for some integers a and b, both G and H admit (a, b)-bipartitions. In this note, we prove that any two compatible trees of order n can be packed into a complete bipartite graph of order at most n + 1. We also provide a family of infinitely many pairs of compatible trees which cannot be packed into a complete bipartite graph of the same order. A theorem about packing two forests into a complete bipartite graph is derived from this result. © 1996 John Wiley & Sons, Inc.     

Packing Arrays and Packing Designs

A packing array is a b × k array, A with entriesa _i,j from a g-ary alphabet such that given any two columns,i and j, and for all ordered pairs of elements from a g-ary alphabet,(g ₁, g ₂), there is at most one row, r, such thata _r,i = g ₁ anda _r,j = g ₂. Further, there is a set of at leastn rows that pairwise differ in each column: they are disjoint. A central question is to determine, forgiven g, k and n, the maximum possible b. We examine the implications whenn is close to g. We give a brief analysis of the case n = g and showthat 2g rows is always achievable whenever more than g exist. We give an upper bound derivedfrom design packing numbers when n = g – 1. When g + 1 k then this bound is always at least as good as the modified Plotkin bound of [12]. When theassociated packing has as many points as blocks and has reasonably uniform replication numbers, we show thatthis bound is tight. In particular, finite geometries imply the existence of a family of optimal or near optimalpacking arrays. When no projective plane exists we present similarly strong results. This article completelydetermines the packing numbers, D(v, k, 1), when .     

Packing triangles in a graph and its complement

How few edge‐disjoint triangles can there be in a graph G on n vertices and in its complement ? This question was posed by P. Erd?s, who noticed that if G is a disjoint union of two complete graphs of order n/2 then this number is n²/12 + o(n²). Erd?s conjectured that any other graph with n vertices together with its complement should also contain at least that many edge‐disjoint triangles. In this paper, we show how to use a fractional relaxation of the above problem to prove that for every graph G of order n, the total number of edge‐disjoint triangles contained in G and is at least n²/13 for all sufficiently large n. This bound improves some earlier results. We discuss a few related questions as well. © 2004 Wiley Periodicals, Inc. J Graph Theory 47: 203–216, 2004     

On the sum of distances along a circle

The arc distance between two points on a circle is their geodesic distance along the circle. We study the sum of the arc distances determined by n points on a circle, which is a useful measure of the evenness of scales and rhythms in music theory. We characterize the configurations with the maximum sum of arc distances by a balanced condition: for each line that goes through the circle center and touches no point, the numbers of points on each side of the line differ by at most one. When the points are restricted to lattice positions on a circle, we show that Toussaint's snap heuristic finds an optimal configuration. We derive closed-form formulas for the maximum sum of arc distances when the points are either allowed to move continuously on the circle or restricted to lattice positions. We also present a linear-time algorithm for computing the sum of arc distances when the points are presorted by the polar coordinates.     

Packing subgraphs in a graph

We show that the problem of packing edges and triangles in a graph in order to cover the maximum number of nodes can be solved in polynomial time. More generally we present results for the problem of packing edges and a family of hypomatchable subgraphs.     

A PTAS for the disk cover problem of geometric objects

We present PTASs for the disk cover problem: given geometric objects and a finite set of disk centers, minimize the total cost for covering those objects with disks under a polynomial cost function on the disks’ radii. We describe the first FPTAS for covering a line segment when the disk centers form a discrete set, and a PTAS for when a set of geometric objects, described by polynomial algebraic inequalities, must be covered. The latter result holds for any dimension.     

The Densest Packing of 19 Congruent Circles in a Circle

Dense packings of n congruent circles in a circle were given by Kravitz in 1967 for n = 2,..., 16. In 1969 Pirl found the optimal packings for n 10, he also conjectured the dense configurations for 11 n 19. In 1994, Melissen provided a proof for n = 11. In this paper we exhibit the densest packing of 19 congruent circles in a circle with the help of a technique developed by Bateman and Erdös.     

Packing two copies of a tree into its third power

A graph H of order n is said to be k-placeable into a graph G of order n, if G contains k edge-disjoint copies of H. It is well known that any non-star tree T of order n is 2-placeable into the complete graph K_n. In the paper by Kheddouci et al. [Packing two copies of a tree into its fourth power, Discrete Math. 213 (2000) 169-178], it is proved that any non-star tree T is 2-placeable into T⁴. In this paper, we prove that any non-star tree T is 2-placeable into T³.     

Dissecting the square into seven or nine congruent parts

We give a computer-based proof of the following fact: If a square is divided into seven or nine convex polygons, congruent among themselves, then the tiles are rectangles. This confirms a new case of a conjecture posed first by Yuan, Zamfirescu and Zamfirescu and later by Rao, Ren and Wang. Our method allows us to explore other variants of this question, for example, we also prove that no rectangle can be tiled by five or seven congruent non-rectangular polygons.     

Packing tripods: Narrowing the density gap

In 1984, Stein and his co-authors posed a problem concerning simple three-dimensional shapes, known as semicrosses, or tripods. By definition, a tripod of order n is formed by a corner and the three adjacent edges of an integer n×n×n cube. How densely can one fill the space with non-overlapping tripods of a given order? In particular, is it possible to fill a constant fraction of the space as tripod order tends to infinity? In this paper, we settle the second question in the negative: the fraction of the space that can be filled with tripods must be infinitely small as the order grows. We also make a step towards the solution of the first question, by improving the currently known asymptotic lower bound on tripod packing density, and by presenting some computational results on low-order packings.     

Efficiently packing unequal disks in a circle

Placing non-overlapping circles in a smallest container is a hard task. In this paper we present our strategy for optimally placing circles in a smallest circle which led us to win an international competition by properly mixing local and global optimization strategies with random search and local moves.     

Packing Triangles in Regular Tournaments

We prove that a regular tournament with n vertices has more than pairwise arc‐disjoint directed triangles. On the other hand, we construct regular tournaments with a feedback arc set of size less than , so these tournaments do not have pairwise arc‐disjoint triangles.     

Mixed-integer programming models for nesting problems

Several industrial problems involve placing objects into a container without overlap, with the goal of minimizing a certain objective function. These problems arise in many industrial fields such as apparel manufacturing, sheet metal layout, shoe manufacturing, VLSI layout, furniture layout, etc., and are known by a variety of names: layout, packing, nesting, loading, placement, marker making, etc. When the 2-dimensional objects to be packed are non-rectangular the problem is known as the nesting problem. The nesting problem is strongly NP-hard. Furthermore, the geometrical aspects of this problem make it really hard to solve in practice. In this paper we describe a Mixed-Integer Programming (MIP) model for the nesting problem based on an earlier proposal of Daniels, Li and Milenkovic, and analyze it computationally. We also introduce a new MIP model for a subproblem arising in the construction of nesting solutions, called the multiple containment problem, and show its potentials in finding improved solutions.