On the discrepancy of strongly unimodular matrices

A (0, 1) matrix A is strongly unimodular if A is totally unimodular and every matrix obtained from A by setting a nonzero entry to 0 is also totally unimodular. Here we consider the linear discrepancy of strongly unimodular matrices. It was proved by Lováz, et.al. [5] that for any matrix A, lindisc(A) ≤ herdisc(A). (1) When A is the incidence matrix of a set-system, a stronger inequality holds: For any family H of subsets of {1, lindisc(H) ≤ (1 − t n)herdisc(H). where t n ≥ 2 −2 n (J. Spencer, [6]). In this paper we prove that the constant t n can be improved to 3 −(n+1)/2 for strongly unimodular matrices.