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