On Maximal Frequent and Minimal Infrequent Sets in Binary Matrices

  • Authors:

  • E. Boros;V. Gurvich;L. Khachiyan;K. Makino

  • Affiliations:

  • RUTCOR, Rutgers University, 640 Bartholomew Road, Piscataway, New Jersey 08854-8003, USA e-mail: boros@rutcor.rutgers.edu;RUTCOR, Rutgers University, 640 Bartholomew Road, Piscataway, New Jersey 08854-8003, USA e-mail: gurvich@rutcor.rutgers.edu;Department of Computer Science, Rutgers University, 110 Frelinghuysen Road, Piscataway, New Jersey 08854-8019, USA e-mail: leonid@cs.rutgers.edu;Division of Systems Science, Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka, 560-8531, Japan e-mail: makino@sys.es.osaka-u.ac.jp

  • Venue:
  • Annals of Mathematics and Artificial Intelligence
  • Year:
  • 2003

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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.