Fooling-sets and rank

Abstract

An n×n matrix M is called a fooling-set matrix of size n if its diagonal entries are nonzero and M k,ℓ M ℓ,k = 0 for every k = ℓ. Dietzfelbinger, Hromkovič, and Schnitger (1996) showed that n ≤ (rk M) 2 , regardless of over which field the rank is computed, and asked whether the exponent on rk M can be improved. We settle this question. In characteristic zero, we construct an infinite family of rational fooling-set matrices with size n = rk M+1 2 . In nonzero characteristic, we construct an infinite family of matrices with n = (1 + o(1))(rk M) 2 .

Rational realization of the minimum ranks of nonnegative sign pattern matrices

“…We point out that the preceding result resolves an open problem posed in [15], as this result implies the existence of a fooling-set submatrix (which includes a nonsingular triangular matrix as a special case, see [15]) of order d + 1 of the nonnegative sign pattern (or the (0, 1)-matrix) determined by the convex polytope K of dimen-sion d, while the previously known lower bound for the order of a largest fooling-set submatrix is √ d. Note that in general, up to permutation equivalence, a nonnegative sign pattern A with minimum rank r may not have a triangular submatrix of order r with all diagonal entries positive. For instance, the sign pattern…”

“…As indicated in the preceding theorems, the study of the minimum ranks of nonnegative sign patterns leads to investigation of convex polytopes. We now show that due to the two-way correspondence between sign pattern matrices and pointhyperplane configurations, a convex polytope also determines a nonnegative sign pattern naturally, and this approach has been exploited previously, see [15]. Theorem 2.9.…”

The (Minimum) Rank of Typical Fooling-Set Matrices

“…It is known that, for fields F with nonzero characteristic, this upper bound is asymptotically attained [6], and for all fields, it is attained up to a multiplicative constant [5]. These results, however, require sophisticated constructions.…”

Fooling sets and the Spanning Tree polytope

“…Most importantly, there can never be a fooling set larger than the square of the dimension of the polytope ( [5]; see also [2,11]). However, since that bound is known to be tight [7,6], and the dimension of the spanning tree polytope is Ω(n 2 ), a fooling set bound of Ω(n 3 ) for the Spanning Tree polytope is possible. A small improvement was made in [12], where a O(n 8/3 log n) upper bound for the largest possible fooling set for the Spanning Tree polytope is shown.…”