Exact Solution of the Knapsack Problem

Hans Kellerer (), Ulrich Pferschy () and David Pisinger ()
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Hans Kellerer: University of Graz, Department of Statistics and Operations Research
Ulrich Pferschy: University of Graz, Department of Statistics and Operations Research
David Pisinger: University of Copenhagen, DIKU, Department of Computer Science

Chapter 5 in Knapsack Problems, 2004, pp 117-160 from Springer

Abstract: Abstract Assume that a set of n items is given, each item j having an integer profit p j and an integer weight w j . The knapsack problem asks to choose a subset of the items such that their overall profit is maximized, while the overall weight does not exceed a given capacity c. Introducing binary variables x j to indicate whether item j is included in the knapsack or not the model may be defined: (5.1) $$ {\rm{(KP)}}\,{\rm{maximize}}\;\sum\limits_{j = 1}^n {{p_j}{x_j}} $$ (5.2) $$ {\rm{subject}}\;{\rm{to}}\;\sum\limits_{j = 1}^n {{w_j}{x_j}} \le c,$$ (5.3) $$ {x_j} \in \left\{ {0,1} \right\},\;j = 1,...,n.$$

Keywords: Dynamic Programming; Knapsack Problem; Cardinality Constraint; Core Problem; Dynamic Programming Recursion (search for similar items in EconPapers)
Date: 2004
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DOI: 10.1007/978-3-540-24777-7_5

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