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Clique-based facets for the precedence constrained knapsack problem

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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.

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References

  1. Achterberg, T.: Constraint Integer Programming. PhD Thesis, Technische Universität, Berlin (2007)

  2. Balas E.: Disjunctive programming. Ann. Discr. Math 5, 3–51 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bley, A.: Routing and Capacity Optimization for IP Networks. PhD Thesis, Technische Universität, Berlin (2007)

  4. Bley A., Boland N., Fricke C., Froyland G.: A strengthened formulation and cutting planes for the open pit mine production scheduling problem. Comput. Oper. Res. 37, 1641–1647 (2010)

    Article  MATH  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 Borndörfer, R., Kormos, Z.: An Algorithm for Maximum Cliques, Unpublished working paper, Konrad-Zuse-Zentrum für Informationstechnik, Berlin (1997)

  7. Boyd E.A.: Polyhedral results for the precedence-constrained knapsack problem. Discret. Appl. Math 41, 185–201 (1993)

    Article  MATH  Google Scholar 

  8. Caccetta L., Hill S.P.: An application of branch and cut to open pit mine scheduling. J. Glob. Optim. 27, 349–365 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  9. Fricke, C.: Applications of Integer programming in Open Pit Mining. PhD Thesis, University of Melbourne (2006)

  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 

  11. Ibarra O.H., Kim C.E.: Approximation algorithms for certain scheduling problems. Math. Oper. Res 4, 197–204 (1978)

    Article  MathSciNet  Google Scholar 

  12. Johnson D.S., Niemi K.A.: On knapsacks, partitions, and a new dynamic programming technique for trees. Math. Oper. Res. 8, 1–14 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  13. Nemhauser G.L., Wolsey L.A.: Integer and Combinatorial Optimization. Wiley-Interscience, New York (1988)

    MATH  Google Scholar 

  14. Park K., Park S.: Lifting cover inequalities for the precedence-constrained knapsack problem. Discret. Appl. Math. 72, 219–241 (1997)

    Article  MATH  Google Scholar 

  15. Shaw D.X., Cho G., Chang H.: A depth-first dynamic programming procedure for the extended tree knapsack problem in local access network design. Telecommun. Syst. 7, 29–43 (1997)

    Article  Google Scholar 

  16. Stecke K.E., Kim I.: A study of part type selection approaches for short-term production planning. Int. J. Flex. Manuf. Syst. 1, 7–29 (1988)

    Article  Google Scholar 

  17. Van de Leensel R.L.M.J., van Hoesel C.P.M., van de Klundert J.J.: Lifting valid inequalities for the precedence constrained knapsack problem. Math. Program. 86, 161–185 (1999)

    Article  MathSciNet  MATH  Google Scholar 

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Correspondence to Natashia Boland.

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Boland, N., Bley, A., Fricke, C. et al. Clique-based facets for the precedence constrained knapsack problem. Math. Program. 133, 481–511 (2012). https://doi.org/10.1007/s10107-010-0438-7

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  • DOI: https://doi.org/10.1007/s10107-010-0438-7

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