We show that the linear discrepancy of a totally
unimodular *m*×*n* matrix *A* is at most

.

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

: 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.

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* A preliminary version of this result appeared at SODA 2001. This work was partially supported by the graduate school ‚Effiziente Algorithmen und Multiskalenmethoden‘, Deutsche Forschungsgemeinschaft

† A similar result has been independently obtained by T. Bohman and R. Holzman and presented at the Conference on Hypergraphs (Gyula O. H. Katona is 60), Budapest, in June 2001.

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### Cite this article

Doerr, B. Linear Discrepancy of Totally
Unimodular Matrices*†.
*Combinatorica* **24, **117–125 (2004). https://doi.org/10.1007/s00493-004-0007-x

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Classification (2000):*

- 11K38
- 90C05
- 05C65