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.

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Correspondence to Benjamin Doerr.

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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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Doerr, B. Linear Discrepancy of Totally Unimodular Matrices*†. Combinatorica 24, 117–125 (2004).

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Mathematics Subject Classification (2000):

  • 11K38
  • 90C05
  • 05C65