Examples of new facets for the precedence constrained knapsack problem

Abstract

We consider the polyhedral structure of the precedence constrained knapsack (PCK) problem, also known as the partially ordered knapsack problem. A set of items N is given, along with a partial order, or set of precedence relationships, on the items, denoted by S ⊆ N × N. A precedence relationship (i, j) ∈ S exists if item i can be placed in the knapsack only if item j is in the knapsack. Each item i ∈ N has a value ci ∈ Z and a weight ai ∈ Z, and the knapsack has a capacity b ∈ Z. The PCK problem seeks a maximum value subset of N whose total weight does not exceed the knapsack capacity, and that also satisfies the precedence relationships. The precedence constraints can be represented by the directed acyclic graph G = (N, S), where the node set is the set of all items N, and each precedence constraint in S is represented by a directed arc. Note that the precedence constraints are transitive, so without loss of generality we assume that S does not contain any redundant relationships, that is, S is the set of all immediate predecessor arcs. If G contains a cycle, all nodes within the cycle must either all be included in, or all be excluded from, the knapsack. Hence the cycle can be contracted into a single node, with cumulative value and weight coefficients, and the resulting directed graph is acyclic. An integer programming formulation of the PCK problem is as follows. Let

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Cite this paper

@inproceedings{Fricke2006ExamplesON, title={Examples of new facets for the precedence constrained knapsack problem}, author={Christopher Fricke}, year={2006} }