Approximating the Geometric Knapsack Problem in Near-Linear Time and Dynamically

One important goal in algorithm design is determining the best running time for solving a problem (approximately). For some problems, we know the optimal running time, assuming certain conditional lower bounds. In this paper, we study the d-dimensional geometric knapsack problem in which we are far from this level of understanding. We are given a set of weighted d-dimensional geometric items like squares, rectangles, or hypercubes and a knapsack which is a square or a (hyper-)cube. Our goal is to select a subset of the given items that fit non-overlappingly inside the knapsack, maximizing the total profit of the packed items. We make a significant step towards determining the best running time for solving these problems approximately by presenting approximation algorithms whose running times are near-linear, i.e., O(n⋅poly(log n)), for any constant d and any parameter ε > 0 (the exponent of log n depends on d and 1/ε). In the case of (hyper)-cubes, we present a (1+ε)-approximation algorithm. This improves drastically upon the currently best known algorithm which is a (1+ε)-approximation algorithm with a running time of n^{O_{ε,d}(1)} where the exponent of n depends exponentially on 1/ε and d. In particular, our algorithm is an efficient polynomial time approximation scheme (EPTAS). Moreover, we present a (2+ε)-approximation algorithm for rectangles in the setting without rotations and a (17/9+ε)≈ 1.89-approximation algorithm if we allow rotations by 90 degrees. The best known polynomial time algorithms for this setting have approximation ratios of 17/9+ε and 1.5+ε, respectively, and running times in which the exponent of n depends exponentially on 1/ε. In addition, we give dynamic algorithms with polylogarithmic query and update times, having the same approximation guarantees as our other algorithms above. Key to our results is a new family of structured packings which we call easily guessable packings. They are flexible enough to guarantee the existence of profitable solutions while providing enough structure so that we can compute these solutions very quickly.