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.

This is a preview of subscription content, access via your institution.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Benjamin Doerr.

Additional information

* 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.

Rights and permissions

Reprints and Permissions

About this article

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

Download citation

Mathematics Subject Classification (2000):

  • 11K38
  • 90C05
  • 05C65