Abstract
Given two finite sets of points X + and X − in \(\mathbb{R}^n\) n, the maximum box problem consists of finding an interval (“box”) B = {x : l ≤ x ≤ u} such that B ∩ X − = ∅, and the cardinality of B ∩ X + is maximized. A simple generalization can be obtained by instead maximizing a weighted sum of the elements of B ∩ X +. While polynomial for any fixed n, the maximum box problem is \( {\mathcal{N}}{\mathcal{P}}\)-hard in general. We construct an efficient branch-and-bound algorithm for this problem and apply it to a standard problem in data analysis. We test this method on nine data sets, seven of which are drawn from the UCI standard machine learning repository.
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Eckstein, J., Hammer, P.L., Liu, Y. et al. The Maximum Box Problem and its Application to Data Analysis. Computational Optimization and Applications 23, 285–298 (2002). https://doi.org/10.1023/A:1020546910706
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DOI: https://doi.org/10.1023/A:1020546910706