In this paper, we study the on-line version of the bin-packing problem. We analyze the approximation behavior of an on-line bin-packing algorithm under an approximation criterion called differential ratio. We are interested in two types of results: the differential competitivity ratio guaranteed by the on-line algorithm and hardness results that account for the difficulty of the problem and for...

We consider the bin packing problem with d different item sizes si and item multiplicities ai, where all numbers are given in binary encoding. This problem formulation is also known as the 1-dimensional cutting stock problem. In this work, we provide an algorithm which, for constant d, solves bin packing in polynomial time. This was an open problem for all d ≥ 3. In fact, for constant d our alg...

We introduce and motivate several variants of the bin packing problem where bins are assigned to time slots, minimum maximum lags required between some pairs items. suggest two integer programming formulations for general problem: a compact one stronger formulation with an exponential number variables constraints. propose branch-cut-and-price approach that exploits latter formulation. For this ...

This work discusses my attempts to extend Kenyon and Mitzenmacher’s technique for proving diveregnce of the online approximation algorithm Best Fit to Random Fit – another approximation algorithm for the well-known NP-hard problem of bin packing. In specific, the paper goes over Kenyon and Mitzenmacher’s recent advances on divergence of the waste of Best Fit bin packing for the skewed distribut...

We present the Generalized Bin Packing Problem (GBPP), a new packing problem where, given a set of items characterized by volume and profit and a set of bins with given volumes and costs, one aims to select the subsets of profitable items and appropriate bins to optimize an objective function which combines the cost of using the bins and the profit yielded by loading the selected items. The GBP...

Background: Bin packing is an NP hard optimization problem of packing items of given sizes into minimum number of capacity limited bins. Besides the basic problem, numerous other variants of bin packing exist. The cardinality constrained bin packing adds an additional constraint that the number of items in a bin must not exceed a given limit Nmax. Objectives: Goal of the paper is to present a p...

We consider the offline and online versions of a bin packing problem called bin packing with conflicts. Given a set of items V = {1, 2, . . . , n} with sizes s1, s2 . . . , sn ∈ [0, 1] and a conflict graph G = (V, E), the goal is to find a partition of the items into independent sets of G, where the total size of each independent set is at most one, so that the number of independent sets in the...