## DIMACS TR: 2000-37

## An inequality limiting the number of maximal frequent sets

### Authors: E. Boros,V. Gurvich, L. Khachiyan and K. Makino

**
ABSTRACT
**

Given an arbitrary $m\times n$ binary matrix $A$ and a threshold
$t \in \{1, \ldots, m\}$, a subset $C$ of the columns is called
$t$-{\em frequent} if there are at least $t$ rows having a $1$ in
each of these columns, and otherwise $C$ is called $t$-{\em
infrequent}. Denoting by $\alpha$ the number of maximal
$t$-frequent, and by $\beta$ the number of minimal $t$-infrequent
column sets in the given matrix $A$, we prove that the inequality
$\alpha \leq (m-t+1)\beta$ holds. This inequality is sharp, and
allows for an incremental quasi-polynomial algorithm for
generating all minimal $t$-infrequent sets, while the analogous
problem for maximal $t$-frequent sets is NP-hard. Our proof is
based on an inequality from extremal set theory, which may be of
independent interest.

Paper Available at:
ftp://dimacs.rutgers.edu/pub/dimacs/TechnicalReports/TechReports/2000/2000-37.ps.gz

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