2D knapsack: Packing squares

Packing the meeting summarization knapsack

Despite considerable work in automatic meeting summarization over the last few years, comparing results remains difficult due to varied task conditions and evaluations. To address this issue, we present a method for determining the best possible extractive summary given an evaluation metric like ROUGE. Our oracle system is based on a knapsack-packing framework, and though NP-Hard, can be solved...

On Packing Squares with Equal Squares

The following problem arises in connection with certain multidimensional stock cutting problems : How many nonoverlapping open unit squares may be packed into a large square of side a? Of course, if a is a positive integer, it is trivial to see that a2 unit squares can be successfully packed . However, if a is not an integer, the problem becomes much more complicated . Intuitively, one feels th...

Translative Packing of Unit Squares into Squares

Packing a Knapsack of Unknown Capacity

We study the problem of packing a knapsack without knowing its capacity. Whenever we attempt to pack an item that does not fit, the item is discarded; if the item fits, we have to include it in the packing. We show that there is always a policy that packs a value within factor 2 of the optimum packing, irrespective of the actual capacity. If all items have unit density, we achieve a factor equa...

On Packing 2D Irregular Shapes

Designing and implementing an intelligent system that tackles the problem of placing two-dimensional shapes on a surface, such that no shapes overlap and the uncovered surface area is minimized, is highly important in industrial applications. However, it is also interesting from the scientific perspective, in terms of artificial intelligence, since autonomous systems developed up to now have fo...