On the supermodular knapsack problem

In this paper we introduce binary knapsack problems where the objective function is nonlinear, and investigate their Lagrangean and continuous relaxations. Some of our results generalize previously known theorems concerning linear and quadratic knapsack problems. We investigate in particular the case in which the objective function is supermodular. Under this hypothesis, although the problem remains NP-hard, we show that its Lagrangean dual and its continuous relaxation can be solved in polynomial time. We also comment on the complexity of recognizing supermodular functions. The particular case in which the knapsack constraint is of the cardinality type is also addressed and some properties of its optimal value as a function of the right hand side are derived.Work done while the authors were visiting Rutgers University.