An approximation algorithm for solving unconstrained two-dimensional knapsack problems

An efficient heuristic for solving two-dimensional knapsack problems is proposed. The algorithm selects an optimal subset of optimal generated strips by solving a sequence of one-dimensional knapsack problems. We show that the number of these knapsacks can be reduced to only four knapsacks. The algorithm gives an excellent worst-case experimental approximation ratio (0.98), and a high percentage of optimal solutions (91%). From this heuristic, we derive an approximation algorithm for which we prove some refined bounds and we show that its approximation ratio is 4/9. Our numerical study on large size instances shows the efficiency of these algorithms for solving real-world problems that are hardly handled by other known methods, which are often limited by computer storage facilities.