Online Removable Square Packing

Abstract

The online removable square packing problem is a two dimensional version of the online removable Knapsack problem. For a sequence of squares with side length at most 1, we are requested to pack a subset of them into a unit square in an online fashion where the online player can decide whether to take the current square or not and which squares currently in the unit square to remove. The goal is to maximize the total packed area. Our results include: (i) Any online algorithm cannot achieve a better competitive ratio than ($\sqrt{5}+3)/2 \approx 2.618$. (ii) The matching upper bound is achieved by a relatively simple online algorithm if repacking is allowed. (iii) Without repacking, we can achieve an upper bound of 3 by using the concept of bricks by Januszewski and Lassak [11]. (iv) The offline version of the problem admits a PTAS.