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Facets of Knapsack Polytopes with Bipartite Precedence Constraints

We consider the precedence-constrained knapsack problem. which is a knapsack problem with precedence constraints imposed on the set of variables. Especially, we focus on the case where the precedence constraints cir be represented as a bipartite graph, which occurs most frequently in applications. Based on the previous studios for the general case, we specialize the polyhedral results on the related polytope and derive stronger results on the facet-defining properties of the inequalities.