We study the following variant of the bin packing problem. We are given a set of items, where each item has a (non-negative) size and a color. We are also given an integer parameter k, and the goal is to partition the items into a minimum number of subsets such that for each subset S in the solution, the total size of the items in S is at most 1 (as in the classical bin packing problem) and the...

We study the online bin packing problem, in which a list of items with integral size between 1 to B arrives one at a time. Each item must be assigned in a bin of capacity B upon its arrival without any information on the next items, and the goal is to minimize the number of used bins. We present an asymptotic competitive scheme, i.e., for any > 0, the asymptotic competitive ratio is at most ρ∗ ...

A point in the d-dimensional integer lattice Z d $\mathbb {Z}^d$ is primitive when its coordinates are relatively prime. Two points multiples of one another they opposite, and for this reason, we consider half within lattice, ones whose first non-zero coordinate positive. We solve packing problem that asks largest possible number such absolute values any given sum to at most a fixed k. present ...