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

Abstract

An 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 work, we study the $d$-dimensional geometric knapsack problem where 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. We want to select a subset of 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 with near-linear running times for any constant dimension d and any constant parameter $\epsilon$. For (hyper)-cubes, we present a $(1+\epsilon )$-approximation algorithm whose running time drastically improves upon the known $(1+\epsilon )$-approximation algorithm which has a running time where the exponent of n depends exponentially on $1/\epsilon$ and $d$. Moreover, we present a $(2+\epsilon )$-approximation algorithm for rectangles in the setting without rotations and a $(17/9+\epsilon )$-approximation algorithm if we allow rotations by 90 degrees. The best known polynomial time algorithms for these settings have approximation ratios of $17/9+\epsilon$ and $1.5+\epsilon$, respectively, and running times in which the exponent of n depends exponentially on $1/\epsilon$. We also give dynamic algorithms with polylogarithmic query and update times and the same approximation guarantees as the 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 profitable solutions and structured enough so that we can compute these solutions quickly.