For a set D of polyominoes, a packing of the plane with D is a maximal set of copies of polyominoes from D that are not overlapping. A packing with smallest density is called a clumsy packing. We give an example of a set D such that any clumsy packing is aperiodic. In addition, we compute the smallest possible density of a clumsy packing when D consists of a single polyomino of a given size and...

For a set D of polyominoes, a packing of the plane with D is a maximal set of copies of polyominoes from D that are not overlapping. A packing with smallest density is called clumsy packing. We give an example of a set D such that any clumsy packing is aperiodic. In addition, we compute the smallest possible density of a clumsy packing when D consist of a single polyomino of a given size and sh...

We study the d-dimensional vector bin packing problem, a well-studied generalization of bin packing arising in resource allocation and scheduling problems. Here we are given a set of d-dimensional vectors v1, . . . , vn in [0, 1] , and the goal is to pack them into the least number of bins so that for each bin B, the sum of the vectors in it is at most 1 in every dimension, i.e., || ∑ vi∈B vi||...

We present a computer code that implements a general Tabu Search technique for the solution of two-and three-dimensional bin packing problems, as well as virtually any of their variants requiring the minimization of the number of bins. The user is only requested to provide a procedure that gives an approximate solution to the actual variant to be solved.

In this paper, we consider the three-dimensional orthogonal bin packing problem, which is a generalization of the well-known bin packing problem. We present new lower bounds for the problem and demonstrate that they improve the best previous results. The asymptotic worst-case performance ratio of the lower bounds is also proved. In addition, we study the non-oriented model, which allows items t...

In this paper an approximation algorithm for the three-dimensional bin packing problem is proposed and its performance bound is investigated. To obtain such a bound a modified bin packing algorithm is considered for a two-dimensional problem with bounded bin and its area utilization is estimated. Finally, a hard example gives a lower bound of the performance bound. '

We show a new algorithm and improved bound for the online square-into-square packing problem using a hybrid shelf-packing approach. This 2-dimensional packing problem involves packing an online sequence of squares into a unit square container without any two squares overlapping. We seek the largest area α such that any set of squares with total area at most α can be packed. We show an algorithm...

We study the Best Fit algorithm for on line bin packing under the distribution in which the item sizes are uniformly distributed in the discrete range f k k j kg Our main result is that in the case j k the asymptotic expected waste remains bounded This settles an open problem of Co man et al and involves a detailed analysis of the in nite multi dimensional Markov chain underlying the algorithm

In [Int. J. Found. Computer Sci. 22 (2011) 1971–1993] the authors introduced a very general problem called Graph-Bin Packing (GBP). It requires a mapping μ : V (G) → V (H) from the vertex set of an input graph G into a fixed host graph H, which, among other conditions, satisfies that for each pair u, v of adjacent vertices the distance of μ(u) and μ(v) in H is between two prescribed bounds. In ...