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+ε [Jansen and Zhang, SODA 2004]. In this article we present a polynomial-time 17/9+ε < 1.89-approximation, which improves to 558/325+ε < 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+ε (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+ε [Jansen and Zhang, SODA 2004]. We present a polynomial-time (3/2+ε)-approximation for this setting, which improves to 4/3+ε in the cardinality case.

We study the strip packing problem, a classical packing problem which generalizes both bin packing and makespan minimization. Here we are given a set of axis-parallel rectangles in the two-dimensional plane and the goal is to pack them in a vertical strip of fixed width such that the height of the obtained packing is minimized. The packing must be non-overlapping and the rectangles cannot be rotated. A reduction from the partition problem shows that no approximation better than 3/2 is possible for strip packing in polynomial time (assuming P!=NP). Nadiradze and Wiese [SODA16] overcame this barrier by presenting a (7/5+epsilon)-approximation algorithm in pseudo-polynomial-time (PPT). As the problem is strongly NP-hard, it does not admit an exact PPT algorithm (though a PPT approximation scheme might exist). In this paper we make further progress on the PPT approximability of strip packing, by presenting a (4/3+epsilon)-approximation algorithm. Our result is based on a non-trivial repacking of some rectangles in the "empty space" left by the construction by Nadiradze and Wiese, and in some sense pushes their approach to its limit. Our PPT algorithm can be adapted to the case where we are allowed to rotate the rectangles by 90 degrees, achieving the same approximation factor and breaking the polynomial-time approximation barrier of 3/2 for the case with rotations as well.