On the supermodular knapsack problem
 G. Gallo,
 B. Simeone
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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 NPhard, 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.
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 Title
 On the supermodular knapsack problem
 Journal

Mathematical Programming
Volume 45, Issue 13 , pp 295309
 Cover Date
 19890801
 DOI
 10.1007/BF01589108
 Print ISSN
 00255610
 Online ISSN
 14364646
 Publisher
 SpringerVerlag
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 Authors

 G. Gallo ^{(1)}
 B. Simeone ^{(2)}
 Author Affiliations

 1. Dipartimento di Informatica, Università di Pisa, 56100, Pisa, Italy
 2. Dipartimento de Statistica, Università di Roma “La Sapienza”, 00185, Rome, Italy