Lattice approximation and linear discrepency of totally unimodular matrices

This paper shows that the lattice approximation problem for totally unimodular matrices <i>A</i> ∈ R^{<i>m</i>×<i>n</i>} can be solved efficiently and optimally via a linear programming approach. The complexity of our algorithm is <i>&Ogr;</i>(log <i>m</i>) times the complexity of finding an extremal point of a polytope in Rⁿ described by 2(<i>m</i> + <i>n</i>) linear constraints. We also consider the worst-case approximability. This quantity is usually called linear discrepancy lindisc(<i>A</i>). For any totally unimodular <i>m</i> × <i>n</i> matrix <i>A</i> we show lindisc(<i>A</i>) ≤ min{1 - 1/<i>n</i>+1, 1 - 1/<i>m</i>}. This bound is sharp. It proves Spencer's conjecture lindisc(<i>A</i>) ≤ (1 - 1/<i>n</i>+1) herdisc(<i>A</i>) for totally unimodular matrices. This seems to be the first time that linear programming is successfully used for a discrepancy problem.