Linear Discrepancy of Totally Unimodular Matrices*?

Abstract

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.