A poset is (r+s)-free if it does not contain two incomparable chains of size r and s, respectively. We prove that when r and s are at least 2, the First-Fit algorithm partitions every (r + s)-free poset P into at most 8(r− 1)(s− 1)w chains, where w is the width of P . This solves an open problem of Bosek, Krawczyk, and Szczypka (SIAM J. Discrete Math., 23(4):1992–1999, 2010).

In this paper, we describe an aircraft loading problem submitted by the French military agency (DGA) as part of a more general military airlift planning problem. It can be viewed as a kind of bi-dimensional bin-packing problem, with heterogeneous bins and several additional constraints. We introduce two-phase methods for solving this NP-hard problem. The first phase consists in building good in...

The two dimensional rectangle packing problem (2RP) is to place a given set of rectangles in the plane without overlap so that the given cost function is minimized. The edges of rectangles are required to be vertical or horizontal. There are various options on the packing rules and objective functions, e.g., rotations are allowed or not, a bin of fixed size to which rectangles should be packed ...

In this paper, we aim at making appear the way Flow and Multicommodity Flow Theory may be used in order to deal with combinatorial geometry problems like the 2D-Bin Packing problem. In order to do it, we introduce a notion of no circuit double flow, we state a Reformulation Theorem which ties some multicommodity flow model with a given bin-packing problem, and we provide an algorithm whose purp...

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...

Bin packing is a well known problem which has a large number of applications. Classical bin packing is a simple model where all bins are identical. In the bin packing problem with variable sized bins, we are given a supply of a variety of sizes. This latter model assumes, however, that the cost of a bin is always defined to be its exact size. In this paper we study the more general problem wher...

In order to verify and test the performance of new packing algorithms relative to existing algorithms, test problems are needed. The scope of published test instances for 2D rectangular and irregular packing has been fairly limited in terms of object size, number and dimension of items. A systematic investigation into the performance of new algorithms requires the use of a range of test problem...

We consider the problem of packing rectangles into bins that are unit squares, where the goal is to minimize the number of bins used. All rectangles have to be packed nonoverlapping and orthogonal, i.e., axis-parallel. We present an algorithm for this problem with an absolute worst-case ratio of 2, which is optimal provided P 6= NP.