We develop a polynomial-time algorithm to optimise a variant of the one-dimensional bin-packing problem with side constraints. We also develop a pseudo-polynomial procedure to actually implement that optimal solution. The speci"c application is the allocation of excess of a population of various types of cards (e.g., left over from a previous selling season) to "xed-sized &&variety packs'' whic...

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.

Y ou could be forgiven for thinking you had misheard her at first: “I always found polymers to be a bit suspicious,” says Tanja Weil, who completed her doctorate at the Max Planck Institute for Polymer Research and is now Director there. Polymers are long, often net-like molecules composed of numerous small chemical subunits that are repeated many times. The highly versatile and durable synthet...

We survey lower bounds for the variant of the two-dimensional bin packing problem where items cannot be rotated. We prove that the dominance relation claimed by Carlier et al.[5] between their lower bounds and those of Boschetti and Mingozzi [1] is not valid. We analyze the performance of lower bounds from the literature and we provide the results of a computational experiment.

We prove that Best Fit bin packing has linear waste on the discrete distributionU{j, k} (where items are drawn uniformly from the set {1/k, 2/k, · · · , j/k}) for sufficiently large k when j = αk and 0.66 ≤ α < 2/3. Our results extend to continuous skewed distributions, where items are drawn uniformly on [0, a], for 0.66 ≤ a < 2/3. This implies that the expected asymptotic performance ratio of ...

We study an on-line bin packing problem. A xed number n of bins, possibly of diierent sizes, are given. The items arrive on-line, and the goal is to pack as many items as possible. It is known that there exists a legal packing of the whole sequence in the n bins. We consider fair algorithms that reject an item, only if it does not t in the empty space of any bin. We show that the competitive ra...