Linear Discrepancy of Totally Unimodular Matrices*?

Abstract

We show that the linear discrepancy of a totally unimodular m × n matrix A is at mostThis bound is sharp. In particular, this result proves Spencer's conjecture lindisc(A) ≤ (1 − 1 n+1 ) herdisc(A) in the special case of totally unimodular matrices. If m ≥ 2, we also show lindisc(A) ≤ 1 − 1 m . Finally we give a characterization of those totally unimodular matrices which have linear discrepancy 1 − 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.

On Geometric Structure of Global Roundings for Graphs and Range Spaces

“…D H ) are integral, and an LP solution automatically gives an IP solution. Thus, a global rounding always exists and can be computed in polynomial time if H is unimodular, and therefore in the literature [2,3,8] unimodular hypergraphs are mainly considered.…”

Unbiased Matrix Rounding

“…Such roundings can be computed efficiently in linear time by a one-pass algorithm resembling Kadane's scanning algorithm (described in Bentley's Programming Pearls [5]). Extensions in different directions have been obtained in [11,12,17,21,23]. This rounding problem has found a number of applications, among others in image processing [1,22].…”

On Properties of a Set of Global Roundings Associated with Clique Connection of Graphs

“…The notion of global rounding on hypergraphs is related to that of discrepancy of hypergraphs [12]. Given a and b 2 ½0; 1 V , we define the discrepancy D H ða; bÞ between them on H by…”