Clique-based facets for the precedence constrained knapsack problem
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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.
KeywordsPrecedence 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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- 1.Achterberg, T.: Constraint Integer Programming. PhD Thesis, Technische Universität, Berlin (2007)Google Scholar
- 3.Bley, A.: Routing and Capacity Optimization for IP Networks. PhD Thesis, Technische Universität, Berlin (2007)Google Scholar
- 5.Bomze I., Budinich M., Pardalos P., Pelillo M.: The maximum clique problem. In: Du, D.-Z., Pardalos, P.M. (eds) Handbook of Combinatorial Optimization, vol 4, Kluwer Academic Press, Dordrecht (1999)Google Scholar
- 6.Borndörfer, R., Kormos, Z.: An Algorithm for Maximum Cliques, Unpublished working paper, Konrad-Zuse-Zentrum für Informationstechnik, Berlin (1997)Google Scholar
- 9.Fricke, C.: Applications of Integer programming in Open Pit Mining. PhD Thesis, University of Melbourne (2006)Google Scholar
- 10.Garey M.R., Johnson D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman and Company, San Francisco (1979)Google Scholar