We introduce the strongly NP-complete pagination problem, an extension of Bin Packing where packing together two items may make them occupy less volume than the sum of their individual sizes. To achieve this property, an item is defined as a finite set of symbols from a given alphabet: while, in Bin Packing, any two such sets would be disjoint, in Pagination, they can share zero, one or more sy...

Cutting stock problems and bin packing problems are basically the same problems. They differ essentially on the variability of the input items. In the first, we have a set of items, each item with a given multiplicity; in the second, we have simply a list of items (each of which we may assume to have multiplicity 1). Many approximation algorithms have been designed for packing problems; a natur...

To the best of our knowledge, this is the first algorithm unifying arbitrary rectilinear block packing and soft block packing. Furthermore, this algorithm handles arbitrary convex or concave rectilinear block packing in the most efficient way compared to other sequence pair-based approaches. At the same time, the algorithm can handle rectangle soft block effectively. The concept of non-redundan...

We present and contrast several efficient two-dimensional packing algorithms for specified aspect ratio. These near-linear algorithms are based on strip packing, tiling, and alternate-bisection methodologies and can be used in the layout of disconnected objects in graph visualization. The parameters that affect the performance of these algorithms as well as the circumstances under which they pe...

We consider a one dimensional storage system where each container can store a bounded amount of capacity as well as a bounded number of items k ≥ 2. This defines the (standard) bin packing problem with cardinality constraints which is an important version of bin packing, introduced by Krause, Shen and Schwetman already in 1975. Following previous work on the unbounded space online problem, we e...

Two-dimensional rectangle packing problem is the problem of packing a series of rectangles into a larger container with maximum area usage of the container. This problem involves many industrial applications, such as shipping, timber cutting, very large scale integration (VLSI) design, etc. It belongs to a subset of classical packing problems and has been shown to be NP hard. For solving this p...

In this paper we are concerned with three lattice problems: the lattice packing problem, the lattice covering problem and the lattice packing-covering problem. One way to find optimal lattices for these problems is to enumerate all finitely many, locally optimal lattices. For the lattice packing problem there are two classical algorithms going back to Minkowski and Voronoi. For the covering and...

We present a novel heuristic for 2D-packing of rectangles inside a rectangular area where the aesthetics of the resulting packing is amenable to generating large collages of photographs or images. The heuristic works by maintaining a sorted collection of vertical segments covering the area to be packed. The segments define the leftmost boundaries of rectangular and possibly overlapping areas th...

case performance bounds for simple one-dimensional packing algorithms. SIAM J. A better algorithm for an ancient scheduling problem. Analysis of several task-scheduling algorithms for a model of multiprogramming computer systems. A new algorithm for on-line bin-packing. 44 43] G. Galambos and J. B. G. Frenk. A simple proof of Liang's lower bound for on-line bin packing and the extension to the ...

The facet of Knapsack ploytope, i.e. convex hull of 0-1 points satisfying a given linear inequality has been presented in this current paper. Such type of facets plays an important role in set covering set partitioning, matroidal-intersection vertex- packing, generalized assignment and other combinatorial problems. Strong covers for facets of Knapsack ploytope has been developed in the first pa...