Mathematical Programming

, Volume 133, Issue 1–2, pp 481–511 | Cite as

Clique-based facets for the precedence constrained knapsack problem

  • Natashia Boland
  • Andreas Bley
  • Christopher Fricke
  • Gary Froyland
  • Renata Sotirov
Full Length Paper Series A

Abstract

We consider a knapsack problem with precedence constraints imposed on pairs of items, known as the precedence constrained knapsack problem (PCKP). This problem has applications in manufacturing and mining, and also appears as a subproblem in decomposition techniques for network design and related problems. We present a new approach for determining facets of the PCKP polyhedron based on clique inequalities. A comparison with existing techniques, that lift knapsack cover inequalities for the PCKP, is also presented. It is shown that the clique-based approach generates facets that cannot be found through the existing cover-based approaches, and that the addition of clique-based inequalities for the PCKP can be computationally beneficial, for both PCKP instances arising in real applications, and applications in which PCKP appears as an embedded structure.

Keywords

Precedence constrained knapsack problem Clique inequalities Integer programming 

Mathematics Subject Classification (2000)

90C10 (integer programming) 90C57 (polyhedral combinatorics, branch-and-bound, branch-and-cut) 

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Copyright information

© Springer and Mathematical Optimization Society 2011

Authors and Affiliations

  • Natashia Boland
    • 1
  • Andreas Bley
    • 2
  • Christopher Fricke
    • 3
  • Gary Froyland
    • 4
  • Renata Sotirov
    • 5
  1. 1.School of Mathematical and Physical SciencesUniversity of NewcastleCallaghanAustralia
  2. 2.Technical University BerlinBerlinGermany
  3. 3.TSG ConsultingMelbourneAustralia
  4. 4.School of Mathematics and StatisticsUniversity of New South WalesSydneyAustralia
  5. 5.Universiteit van Tilburg Warandelaan 2TilburgThe Netherlands

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