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On Maximal Frequent and Minimal Infrequent Sets in Binary Matrices

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Abstract

Given an m×n binary matrix A, a subset C of the columns is called t-frequent if there are at least t rows in A in which all entries belonging to C are non-zero. Let us denote by α the number of maximal t-frequent sets of A, and let β denote the number of those minimal column subsets of A which are not t-frequent (so called t-infrequent sets). We prove that the inequality α≤(m−t+1)β holds for any binary matrix A in which not all column subsets are t-frequent. This inequality is sharp, and allows for an incremental quasi-polynomial algorithm for generating all minimal t-infrequent sets. We also prove that the analogous generation problem for maximal t-frequent sets is NP-hard. Finally, we discuss the complexity of generating closed frequent sets and some other related problems.

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Authors and Affiliations

  1. RUTCOR, Rutgers University, 640 Bartholomew Road, Piscataway, New Jersey, 08854-8003, USA

    E. Boros & V. Gurvich

  2. Department of Computer Science, Rutgers University, 110 Frelinghuysen Road, Piscataway, New Jersey, 08854-8019, USA

    L. Khachiyan

  3. Division of Systems Science, Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka, 560-8531, Japan

    K. Makino

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  1. E. Boros
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  2. V. Gurvich
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  3. L. Khachiyan
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  4. K. Makino
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Boros, E., Gurvich, V., Khachiyan, L. et al. On Maximal Frequent and Minimal Infrequent Sets in Binary Matrices. Annals of Mathematics and Artificial Intelligence 39, 211–221 (2003). https://doi.org/10.1023/A:1024605820527

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  • DOI: https://doi.org/10.1023/A:1024605820527

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