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.