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Maximization of submodular functions: Theory and enumeration algorithms

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  • Goldengorin, Boris

Abstract

Submodular functions are powerful tools to model and solve either to optimality or approximately many operational research problems including problems defined on graphs. After reviewing some long-standing theoretical results about the structure of local and global maxima of submodular functions, Cherenin's selection rules and his Dichotomy Algorithm, we revise the above mentioned theory and show that our revision is useful for creating new non-binary branching algorithms and finding either approximation solutions with guaranteed accuracy or exact ones.

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  • Goldengorin, Boris, 2009. "Maximization of submodular functions: Theory and enumeration algorithms," European Journal of Operational Research, Elsevier, vol. 198(1), pages 102-112, October.
    • Handle: RePEc:eee:ejores:v:198:y:2009:i:1:p:102-112
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    1. G. L. Nemhauser & L. A. Wolsey, 1978. "Best Algorithms for Approximating the Maximum of a Submodular Set Function," Mathematics of Operations Research, INFORMS, vol. 3(3), pages 177-188, August.
    2. Nemhauser, G.L. & Wolsey, L.A., 1978. "Best algorithms for approximating the maximum of a submodular set function," LIDAM Reprints CORE 343, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    3. Boris Goldengorin & Gerard Sierksma & Gert A. Tijssen & Michael Tso, 1999. "The Data-Correcting Algorithm for the Minimization of Supermodular Functions," Management Science, INFORMS, vol. 45(11), pages 1539-1551, November.
    4. A.A. Ageev & M.I. Sviridenko, 2004. "Pipage Rounding: A New Method of Constructing Algorithms with Proven Performance Guarantee," Journal of Combinatorial Optimization, Springer, vol. 8(3), pages 307-328, September.
    5. Egon Balas & Manfred W. Padberg, 1972. "On the Set-Covering Problem," Operations Research, INFORMS, vol. 20(6), pages 1152-1161, December.
    6. Goldengorin, Boris & Tijssen, Gert A. & Tso, Michael, 1999. "The maximization of submodular functions : old and new proofs for the correctness of the dichotomy algorithm," Research Report 99A17, University of Groningen, Research Institute SOM (Systems, Organisations and Management).
    7. Chun-Wa Ko & Jon Lee & Maurice Queyranne, 1995. "An Exact Algorithm for Maximum Entropy Sampling," Operations Research, INFORMS, vol. 43(4), pages 684-691, August.
    8. repec:dgr:rugsom:99a17 is not listed on IDEAS
    9. M. A. Efroymson & T. L. Ray, 1966. "A Branch-Bound Algorithm for Plant Location," Operations Research, INFORMS, vol. 14(3), pages 361-368, June.
    10. Francisco Barahona & Martin Grötschel & Michael Jünger & Gerhard Reinelt, 1988. "An Application of Combinatorial Optimization to Statistical Physics and Circuit Layout Design," Operations Research, INFORMS, vol. 36(3), pages 493-513, June.
    11. Beasley, J. E., 1993. "Lagrangean heuristics for location problems," European Journal of Operational Research, Elsevier, vol. 65(3), pages 383-399, March.
    12. Fisher, M.L. & Nemhauser, G.L. & Wolsey, L.A., 1978. "An analysis of approximations for maximizing submodular set functions," LIDAM Reprints CORE 341, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    13. Jon Lee, 1998. "Constrained Maximum-Entropy Sampling," Operations Research, INFORMS, vol. 46(5), pages 655-664, October.
    14. Lee, Heesang & Nemhauser, George L. & Wang, Yinhua, 1996. "Maximizing a submodular function by integer programming: Polyhedral results for the quadratic case," European Journal of Operational Research, Elsevier, vol. 94(1), pages 154-166, October.
    15. Alain Alcouffe & Gilles Muratet, 1976. "Optimal Location of Plants," Management Science, INFORMS, vol. 23(3), pages 267-274, November.
    16. Stefano Benati, 2003. "An Improved Branch & Bound Method for the Uncapacitated Competitive Location Problem," Annals of Operations Research, Springer, vol. 122(1), pages 43-58, September.
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    8. Francisco Casas & Claudio E. Torres & Ignacio Araya, 2022. "A heuristic search based on diversity for solving combinatorial problems," Journal of Heuristics, Springer, vol. 28(3), pages 287-328, June.

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