The Knapsack Problem with Special Neighbor Constraints on Directed Co-graphs

Abstract

The knapsack problem is one of the best known and most fundamental NP-hard problems in combinatorial optimization. We consider two knapsack problems which contain additional constraints in the form of directed graphs whose vertex set corresponds to the item set. In the 1-neighbor knapsack problem, an item can be chosen only if at least one of its successors is chosen. In the all-neighbors knapsack problem, an item can be chosen only if all of its successors are chosen. For both problems, we consider uniform and general profits and weights. Since all these problems generalize the knapsack problem, they are NP-hard. This motivates us to consider the problem on special graph classes. Therefore, we restrict these problems to directed co-graphs, i.e., directed complement reducible graphs, that are precisely those digraphs which can be defined from the single vertex graph by applying the disjoint union, order and series composition. We show polynomial time solutions for the uniform problems on directed co-graphs and pseudo-polynomial time solutions for the general problems on directed co-graphs. These results improve known worst-case runtimes in comparison to constraints given by unrestricted digraphs.