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On the Set-Covering Problem

Author

Listed:
  • Egon Balas

    (Carnegie-Mellon University, Pittsburgh, Pennsylvania)

  • Manfred W. Padberg

    (Carnegie-Mellon University, Pittsburgh, Pennsylvania)

Abstract

This paper establishes some useful properties of the equality-constrained set-covering problem P and the associated linear program P ′. First, the Dantzig property of transportation matrices is shown to hold for a more general class of matrices arising in connection with adjacent integer solutions to P ′. Next, we show that, for every feasible integer basis to P ′, there are at least as many adjacent feasible integer bases as there are nonbasic columns. Finally, given any two basic feasible integer solutions x 1 and x 2 to P ′, x 2 can be obtained from x 1 by a sequence of p pivots (where p is the number of indices j ϵ N for which x j 1 is nonbasic and x j 2 = 1), such that each solution in the associated sequence is feasible and integer. Some of our results have been conjectured earlier by A ndrew , H offmann , and K rabek in a paper presented to ORSA in 1968.

Suggested Citation

  • Egon Balas & Manfred W. Padberg, 1972. "On the Set-Covering Problem," Operations Research, INFORMS, vol. 20(6), pages 1152-1161, December.
    • Handle: RePEc:inm:oropre:v:20:y:1972:i:6:p:1152-1161
    DOI: 10.1287/opre.20.6.1152
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    Cited by:

    1. Adil Tahir & Guy Desaulniers & Issmail El Hallaoui, 2022. "Integral Column Generation for Set Partitioning Problems with Side Constraints," INFORMS Journal on Computing, INFORMS, vol. 34(4), pages 2313-2331, July.
    2. Rönnberg, Elina & Larsson, Torbjörn, 2014. "All-integer column generation for set partitioning: Basic principles and extensions," European Journal of Operational Research, Elsevier, vol. 233(3), pages 529-538.
    3. Aardal, K.I. & van Hoesel, S., 1995. "Polyhedral Techniques in Combinatorial Optimization," Other publications TiSEM ed028a07-eb6a-4c8d-8f21-d, Tilburg University, School of Economics and Management.
    4. Goldengorin, Boris, 2009. "Maximization of submodular functions: Theory and enumeration algorithms," European Journal of Operational Research, Elsevier, vol. 198(1), pages 102-112, October.
    5. Fernández, Elena & Pozo, Miguel A. & Puerto, Justo & Scozzari, Andrea, 2017. "Ordered Weighted Average optimization in Multiobjective Spanning Tree Problem," European Journal of Operational Research, Elsevier, vol. 260(3), pages 886-903.
    6. Elena Fernández & Justo Puerto & Antonio Rodríguez-Chía, 2013. "On discrete optimization with ordering," Annals of Operations Research, Springer, vol. 207(1), pages 83-96, August.
    7. Adil Tahir & Guy Desaulniers & Issmail El Hallaoui, 2019. "Integral column generation for the set partitioning problem," EURO Journal on Transportation and Logistics, Springer;EURO - The Association of European Operational Research Societies, vol. 8(5), pages 713-744, December.
    8. Rönnberg, Elina & Larsson, Torbjörn, 2009. "Column generation in the integral simplex method," European Journal of Operational Research, Elsevier, vol. 192(1), pages 333-342, January.
    9. Sujeevraja Sanjeevi & Saravanan Venkatachalam, 2021. "Robust flight schedules with stochastic programming," Annals of Operations Research, Springer, vol. 305(1), pages 403-421, October.
    10. Abdelouahab Zaghrouti & François Soumis & Issmail El Hallaoui, 2014. "Integral Simplex Using Decomposition for the Set Partitioning Problem," Operations Research, INFORMS, vol. 62(2), pages 435-449, April.
    11. Abdelouahab Zaghrouti & Issmail El Hallaoui & François Soumis, 2018. "Improved integral simplex using decomposition for the set partitioning problem," EURO Journal on Computational Optimization, Springer;EURO - The Association of European Operational Research Societies, vol. 6(2), pages 185-206, June.
    12. Aardal, K.I. & van Hoesel, S., 1995. "Polyhedral Techniques in Combinatorial Optimization," Discussion Paper 1995-57, Tilburg University, Center for Economic Research.
    13. Aardal, K. & van Hoesel, C.P.M., 1995. "Polyhedral techniques in combinatorial optimization," Research Memorandum 014, Maastricht University, Maastricht Research School of Economics of Technology and Organization (METEOR).
    14. Abdelouahab Zaghrouti & Issmail El Hallaoui & François Soumis, 2020. "Improving set partitioning problem solutions by zooming around an improving direction," Annals of Operations Research, Springer, vol. 284(2), pages 645-671, January.

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