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Cover and Pack Inequalities for (Mixed) Integer Programming

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  • Alper Atamtürk

Abstract

We review strong inequalities for fundamental knapsack relaxations of (mixed) integer programs. These relaxations are the 0-1 knapsack set, the mixed 0-1 knapsack set, the integer knapsack set, and the mixed integer knapsack set. Our aim is to give a unified presentation of the inequalities based on covers and packs and highlight the connections among them. The focus of the paper is on recent research on the use of superadditive functions for the analysis of knapsack polyhedra. We also present some new results on integer knapsacks. In particular, we give an integer version of the cover inequalities and describe a necessary and sufficient facet condition for them. This condition generalizes the well-known facet condition of minimality of covers for 0-1 knapsacks. Copyright Springer Science + Business Media, Inc. 2005

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  • Alper Atamtürk, 2005. "Cover and Pack Inequalities for (Mixed) Integer Programming," Annals of Operations Research, Springer, vol. 139(1), pages 21-38, October.
    • Handle: RePEc:spr:annopr:v:139:y:2005:i:1:p:21-38:10.1007/s10479-005-3442-1
    DOI: 10.1007/s10479-005-3442-1
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    1. Pochet, Y. & Wolsey, L. A., 1995. "Integer knapsack and flow covers with divisible coefficients: polyhedra, optimization and separation," LIDAM Reprints CORE 1149, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    2. Eitan Zemel, 1989. "Easily Computable Facets of the Knapsack Polytope," Mathematics of Operations Research, INFORMS, vol. 14(4), pages 760-764, November.
    3. Laurence A. Wolsey, 1976. "Technical Note—Facets and Strong Valid Inequalities for Integer Programs," Operations Research, INFORMS, vol. 24(2), pages 367-372, April.
    4. Zonghao Gu & George L. Nemhauser & Martin W. P. Savelsbergh, 1999. "Lifted Cover Inequalities for 0-1 Integer Programs: Complexity," INFORMS Journal on Computing, INFORMS, vol. 11(1), pages 117-123, February.
    5. LOUVEAUX, Quentin & WOLSEY, Laurence A., 2003. "Lifting, superadditivity, mixed integer rounding and single node flow sets revisited," LIDAM Reprints CORE 1659, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    6. MARCHAND, Hugues & WOLSEY, Laurence A., 1999. "The 0-1 Knapsack problem with a single continuous variable," LIDAM Reprints CORE 1390, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    7. Alper Atamtürk, 2004. "Sequence Independent Lifting for Mixed-Integer Programming," Operations Research, INFORMS, vol. 52(3), pages 487-490, June.
    8. Zonghao Gu & George L. Nemhauser & Martin W. P. Savelsbergh, 1998. "Lifted Cover Inequalities for 0-1 Integer Programs: Computation," INFORMS Journal on Computing, INFORMS, vol. 10(4), pages 427-437, November.
    9. Hugues Marchand & Laurence A. Wolsey, 2001. "Aggregation and Mixed Integer Rounding to Solve MIPs," Operations Research, INFORMS, vol. 49(3), pages 363-371, June.
    10. Robert Weismantel, 1996. "Hilbert Bases and the Facets of Special Knapsack Polytopes," Mathematics of Operations Research, INFORMS, vol. 21(4), pages 886-904, November.
    11. WOLSEY, Laurence A., 1975. "Faces for a linear inequality in 0-1 variables," LIDAM Reprints CORE 218, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    12. NEMHAUSER, George L. & WOLSEY, Laurence A., 1990. "A recursive procedure to generate all cuts for 0-1 mixed integer programs," LIDAM Reprints CORE 894, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    13. LOUVEAUX, Quentin & WOLSEY, Laurence, 2003. "Lifting, superadditivity, mixed integer rounding and single node flow sets revisited," LIDAM Discussion Papers CORE 2003001, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    14. MARCHAND, Hugues & WOLSEY, Laurence A., 2001. "Aggregation and mixed integer rounding to solve mips," LIDAM Reprints CORE 1513, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    15. CERIA, Sebastian & CORDIER, Cécile & MARCHAND, Hugues & WOLSEY, Laurence A., 1998. "Cutting planes for integer programs with general integer variables," LIDAM Reprints CORE 1334, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    16. POCHET, Yves & WEISMANTEL, Robert, 1998. "The sequential knapsack polytope," LIDAM Reprints CORE 1313, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    17. WOLSEY, Laurence A., 1976. "Facets and strong valid inequalities for integer programs," LIDAM Reprints CORE 246, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
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    1. Kaparis, Konstantinos & Letchford, Adam N., 2008. "Local and global lifted cover inequalities for the 0-1 multidimensional knapsack problem," European Journal of Operational Research, Elsevier, vol. 186(1), pages 91-103, April.

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