On the discrepancy of strongly unimodular matrices

Abstract

A (0,1) matrix A is strongly unimodular if A is totally unimodular and every matrix obtained from A by setting a non-zero entry to 0 is also totally unimodular. Here we consider the linear discrepancy of strongly unimodular matrices. It was proved by Lováz et al. (J. Combin. 7 (1986) 151–160) that for any matrix A, (1) lindisc(A)⩽ herdisc(A). When A is the incidence matrix of a set-system, a stronger inequality holds: For any family H of subsets of {1,2,…,n}, lindisc( H)⩽(1−t n) herdisc( H), where t n⩾2 −2 n (Spencer, Ten Lectures on the Probabilistric Method, 2nd Edition, CBMS-NSF Regional Conferences Series in Applied Mathematics, 1994). In this paper we prove that the constant t n can be improved to 3 −(n+1)/2 for strongly unimodular matrices.