Abstract
This paper describes Benders decomposition approaches to optimally solve set covering problems “ almost” satisfying the consecutive ones property. The decompositions are based on the fact that set covering problems with consecutive ones property have totally unimodular coefficient matrices. Given a binary matrix, a totally unimodular matrix is enforced by filling up every row with ones between its first and its last non-zero entries. The resulting mistake is handled by introducing additional integer variables whose number depends on the reordering of the columns of the given matrix. This leads us to consider the consecutive block minimization problem. Two cutting plane algorithms are proposed and run on a large set of benchmark instances. The results obtained show that the cutting plane algorithms outperform an existing tree search method designed exclusively for such instances.
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Acknowledgments
We are very grateful to N. Ruf, from the university of Kaiserslautern, Germany, for providing us with the data set “ C1P-data”. Also, we would like to acknowledge the anonymous referees for their contribution in improving the quality of the manuscript.
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Haddadi, S. Benders decomposition for set covering problems. J Comb Optim 33, 60–80 (2017). https://doi.org/10.1007/s10878-015-9935-1
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DOI: https://doi.org/10.1007/s10878-015-9935-1