In a recent paper, Brusco, Khn and Steinley [Ann. Oper. Res. 206:611-626 (2013)] conjecture that the 2 bins special case of the one-dimensional minimax bin-packing problem with bin size constraintsmight be solvable in polynomial time. In this note, we show that this problem is NP-hard for the special caseand thatit is strongly NP-hard for the general problem. We also propose a pseudo-polynomial...

Packing boxes into containers is a daily process in many different fields and especially in transport. However, the particular case of air transport brings some new constraints such as the stability or the fragility of the cargo. The distribution of the weight has also to be considered. Moreover, this special case also brings some data such as the dimensions of the possible containers, called U...

Bin packing is a well-known NP-Hard optimization problem which has several real applications. Classical bin packing (BPP) is a simple model where all bins are identical. However, the variable sized bin packing problem (VSBPP) is a generalization of the bin packing problem where bins of different capacities are available for packing a set of items having different weights. The objective is to pa...

Given bins of size B, non-negative values d and ∆, and a list L of items, each item e ∈ L with size se and class ce, we define a shelf as a subset of items packed inside a bin with total items size at most ∆ such that all items in this shelf have the same class. Two subsequent shelves must be separated by a shelf division of size d. The size of a shelf is the total size of its items plus the si...

1. Introduction We consider the following two dimensional bin packing problem: Given a set of rectangular pieces, find a way to pack the pieces into a bin of width 1, bounded below but not above, so as to minimize the height to which the pieces fill the bin. The pieces are not allowed to rotate (not even 90 degrees) or overlap. This problem was first considered in [ 13, which includes several a...

Bin-completion, a bin-oriented branch-and-bound approach, was recently shown to be promising for the bin packing problem. We propose several improvements to bin-completion that significantly improves search efficiency. We also show the generality of bin-completion for packing and covering problems involving multiple containers, and present bin-completion algorithms for the multiple knapsack, bi...

We consider a game-theoretical problem called selfish 2-dimensional bin packing game, a generalization of the 1-dimensional case already treated in the literature. In this game, the items to be packed are rectangles, and the bins are unit squares. The game starts with a set of items arbitrarily packed in bins. The cost of an item is defined as the ratio between its area and the total occupied a...

Cardinality constrained bin packing or bin packing with cardinality constraints is a basic bin packing problem. In the online version with the parameter k ≥ 2, items having sizes in (0, 1] associated with them are presented one by one to be packed into unit capacity bins, such that the capacities of bins are not exceeded, and no bin receives more than k items. We resolve the online problem in t...

We study the multi-dimensional version of the bin packing problem with conflicts. We are given a set of squares V = {1, 2, . . . , n} with sides s1, s2, . . . , sn ∈ [0, 1] and a conflict graph G = (V, E). We seek to find a partition of the items into independent sets of G, where each independent set can be packed into a unit square bin, such that no two squares packed together in one bin overl...