Approximating Geometric Knapsack via L-Packings

We study the two-dimensional geometric knapsack problem (2DK) 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+ ε [Jansen and Zhang, SODA 2004]. In this paper we break the 2 approximation barrier, achieving a polynomialtime 17/9 + ε <; 1.89 approximation, which improves to 558/325 + ε <; 1.72 in the cardinality case. Essentially all prior work on 2DK approximation packs items inside a constant number of rectangular containers, where items inside each container are packed using a simple greedy strategy. We deviate for the first time from this setting: we show that there exists a large profit solution where items are packed inside a constant number of containers plus one L-shaped region at the boundary of the knapsack which contains items that are high and narrow and items that are wide and thin. The items of these two types possibly interact in a complex manner at the corner of the L. The above structural result is not enough however: the best-known approximation ratio for the subproblem in the L-shaped region is 2 + ε (obtained via a trivial reduction to one-dimensional knapsack by considering tall or wide items only). Indeed this is one of the simplest special settings of the problem for which this is the best known approximation factor. As a second major, and the main algorithmic contribution of this paper, we present a PTAS for this case. We believe that this will turn out to be useful in future work in geometric packing problems. We also consider the variant of the problem with rotations (2DKR), where items can be rotated by 90 degrees. Also in this case the best-known polynomial-time approximation factor (even for the cardinality case) is 2+ε[Jansen and Zhang, SODA 2004]. Exploiting part of the machinery developed for 2DK plus a few additional ideas, we obtain a polynomial-time 3/2 + ε-approximation for 2DKR, which improves to 4/3 + ε in the cardinality case.