Greedy algorithms for a class of knapsack problems with binary weights

In this article we identify a class of two-dimensional knapsack problems with binary weights and related three-criteria unconstrained combinatorial optimization problems that can be solved in polynomial time by greedy algorithms. Starting from the knapsack problem with two equality constraints we show that this problem can be solved efficiently by using an appropriate partitioning of the items with respect to their binary weights. Based on the results for this problem we derive an algorithm for the three-criteria unconstrained combinatorial optimization problem with two binary objectives that explores the connectedness of the set of efficient knapsacks with respect to a combinatorial definition of adjacency. Furthermore, we prove that our approach is asymptotically optimal and provide extensive computational experiments that shows that we can solve the three-criteria problem with up to one million items in less than half an hour. Finally, we derive an efficient algorithm for the two-dimensional knapsack problems with binary constraints that only takes into account the results we obtained for the unconstrained three-criteria problem with binary weights.