Using machine learning to predict the number of alternative solutions to a minimum cardinality set covering problem

Brooks Emerick, Yun Lu, Francis J. Vasko


Although the characterization of alternative optimal solutions for linear programming problems is well known, such characterizations for combinatorial optimization problems are essentially non-existent. This is the first article to qualitatively predict the number of alternative optima for a classic NP-hard combinatorial optimization problem, namely, the minimum cardinality (also called unicost) set covering problem (MCSCP). For the MCSCP, a set must be covered by a minimum number of subsets selected from a specified collection of subsets of the given set. The MCSCP has numerous industrial applications that require that a secondary objective is optimized once the size of a minimum cover has been determined. To optimize the secondary objective, the number of MCSCP solutions is optimized. In this article, for the first time, a machine learning methodology is presented to generate categorical regression trees to predict, qualitatively (extra-small, small, medium, large, or extra-large), the number of solutions to an MCSCP. Within the machine learning toolbox of MATLAB®, 600,000 unique random MCSCPs were generated and used to construct regression trees. The prediction quality of these regression trees was tested on 5000 different MCSCPs. For the 5-output model, the average accuracy of being at most one off from the predicted category was 94.2%. 


Minimum cardinality set covering problem; Unicost set covering problem; Machine learning; Regression trees; Number of alternative optimal solution

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International Journal of Industrial Optimization (IJIO)
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