Perfect matrix

In mathematics, a perfect matrix is an m-by-n binary matrix that has no possible k-by-k submatrix K that satisfies the following conditions:^[1]

• k > 3
• the row and column sums of K are each equal to b, where b ≥ 2
• there exists no row of the (m − k)-by-k submatrix formed by the rows not included in K with a row sum greater than b.

The following is an example of a K submatrix where k = 5 and b = 2:

$\begin{bmatrix} 1 & 1 & 0 & 0 & 0 \\ 0 & 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 1 & 1 \\ 1 & 0 & 0 & 0 & 1 \end{bmatrix}.$

References

1. D. M. Ryan, B. A. Foster, An Integer Programming Approach to Scheduling, p.274, University of Auckland, 1981.