We study dense packings of a large number congruent non-overlapping circles inside square by looking for configurations which maximize the packing density, defined as ratio between area occupied disks and container. The search these is carried out with help two algorithms that we have devised: first algorithm in charge obtaining sufficiently starting from random guess, while second improves obt...

This paper proves a bottom-left placement theorem for the rectangle packing problem, stating that if it is possible to orthogonally place n arbitrarily given rectangles into a rectangular container without overlapping, then we can achieve a feasible packing by successively placing a rectangle onto a bottom-left corner in the container. This theorem shows that even for the real-parameter rectang...

The bin packing problem is a of finding an assignment sequence items to minimum number bins, each capacity one. An online algorithm for the that irrevocably assigns item one by from head sequence. Gutin, Jensen, and Yeo (2006) considered version in which all are only two different sizes knows possible advance, gave optimal case when larger size exceeds 1/2. In this paper we provide some cases a...

We prove new lower bounds for suitable competitive ratio measures of two relaxed online packing problems: removable multiple knapsack, and a recently introduced minimum peak appointment scheduling problem. The high level objective in both problems is to pack arriving items sizes at most 1 into bins capacity as efficiently possible, but the exact formalizations differ. In problem, every item has...