Linear Discrepancy of Totally Unimodular Matrices*†

We show that the linear discrepancy of a totally unimodular m×n matrix A is at most $$ {\text{lindisc}}{\left( A \right)} \leqslant 1 - \frac{1} {{n + 1}} $$ .This bound is sharp. In particular, this result proves Spencer’s conjecture $$ {\text{lindisc}}(A) \leqslant {\left( {1 - \frac{1} {{n + 1}}} \right)} $$herdisc(A) in the special case of totally unimodular matrices. If m≥2, we also show $$ {\text{lindisc}}{\left( A \right)} \leqslant 1 - \frac{1} {m} $$.Finally we give a characterization of those totally unimodular matrices which have linear discrepancy $$ 1 - \frac{1} {{n + 1}} $$: Besides m×1 matrices containing a single non-zero entry, they are exactly the ones which contain n+1 rows such that each n thereof are linearly independent. A central proof idea is the use of linear programs.