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