Abstract
The knapsack problem is one of the simplest 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 one-neighbor knapsack problem, an item can be chosen only if at least one of its neighbors is chosen. In the all-neighbors knapsack problem, an item can be chosen only if all its neighbors are chosen. For both problems, we consider uniform and general profits and weights. We prove upper bounds for the time complexity of these problems when restricting the graph constraints to special sets of digraphs. We discuss directed co-graphs, minimal series-parallel digraphs, and directed trees.
Highlights
In recent years, the interest in knapsack problems related to graphs has grown strongly.In addition to an input to a knapsack problem, these problems contain additional constraints in the form of graphs whose vertex set corresponds to the item set of the knapsack problem
In the Conclusions section we show that knapsack with one-neighbor constraint and knapsack with all-neighbors constraint are pseudo-polynomial subset selection problems if zero profits are excluded
We discuss solutions for the one-neighbor and allneighbors knapsack problems for which the input graph is restricted to directed co-graphs, msp-digraphs, and directed trees
Summary
The interest in knapsack problems related to graphs has grown strongly. The partially ordered knapsack problem is to find a subset A ⊆ A which is closed under predecessor, such that A maximizes the profit and fulfills the capacity constraint (1). KP1N can be applied to solve the knapsack problem with forcing graph (KFG): In the KFG problem, for all edges of an undirected forcing graph (A, E), at least one of its two vertices has to be chosen as an item into a feasible KP solution, see (Pferschy and Schauer 2017). The digraph constraint here is that if all predecessors of a vertex v are in a feasible solution, v must be in this solution Bounds for this problem on directed co-graphs and msp-digraphs are proved in Gurski et al (2020a, b).
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