2D Knapsack: Packing Squares

In this paper, we study a two-dimensional knapsack problem: packing squares as many as possible into a unit square. Our results are the following: (i) first, we propose an algorithm called IHS(Increasing Height Shelf), and prove that the packing is optimal if there are at most 5 squares packed in an optimal packing, and this upper bound 5 is sharp; (ii) secondly, if all the items have size(side length) at most 1/k, where k ≥ 1 is a constant number, we propose a simple algorithm with an approximation ratio k2+3k+2/k2/k2 in time O(n log n). (iii) finally, we give a PTAS for the general case, and our algorithm is much simpler than the previous approach[16].