Approximating Geometric Knapsack via L-packings

We study the two-dimensional geometric knapsack problem, in which we are given a set of n axis-aligned rectangular items, each one with an associated profit, and an axis-aligned square knapsack. The goal is to find a (non-overlapping) packing of a maximum profit subset of items inside the knapsack (without rotating items). The best-known polynomial-time approximation factor for this problem (even just in the cardinality case) is 2+e [Jansen and Zhang, SODA 2004]. In this article we present a polynomial-time 17/9+e < 1.89-approximation, which improves to 558/325+e < 1.72 in the cardinality case. Prior results pack items into a constant number of rectangular containers that are filled via greedy strategies. We deviate from this setting and show that there exists a large profit solution where items are packed into a constant number of containers plus one L-shaped region at the boundary of the knapsack containing narrow-high items and thin-wide items. These items may interact in complex manners at the corner of the L. The best-known approximation ratio for the subproblem in the L-shaped region is 2+e (via a trivial reduction to one-dimensional knapsack); hence, as a second major result we present a PTAS for this case that we believe might be of broader utility. We also consider the variant with rotations, where items can be rotated by 90 degrees. Again, the best-known polynomial-time approximation factor (even for the cardinality case) is 2+e [Jansen and Zhang, SODA 2004]. We present a polynomial-time (3/2+e)-approximation for this setting, which improves to 4/3+e in the cardinality case.