Clique-based facets for the precedence constrained knapsack problem

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. N. Boland (B) School of Mathematical and Physical Sciences, University of Newcastle, Callaghan, NSW 2308, Australia e-mail: [email protected] A. Bley Technical University Berlin, Straße des 17. Juni 136, 10623 Berlin, Germany e-mail: [email protected] C. Fricke TSG Consulting, Level 11, 350 Collins Street, Melbourne, VIC 3000, Australia G. Froyland School of Mathematics and Statistics, University of New South Wales, Sydney, NSW 2052, Australia e-mail: [email protected] R. Sotirov Universiteit van Tilburg Warandelaan 2, P.O. Box 90153, 5000 LE Tilburg, The Netherlands e-mail: [email protected]