Discrepancy of Set-systems and Matrices

Lecture 11 . Discrepancy of Set Systems

Discrepancy of Matrices of Zeros and Ones

Let m and n be positive integers, and let R = (r1, . . . , rm) and S = (s1, . . . , sn) be non-negative integral vectors. Let A(R, S) be the set of all m× n (0, 1)-matrices with row sum vector R and column vector S, and let Ā be the m × n (0, 1)-matrix where for each i, 1 ≤ i ≤ m, row i consists of ri 1’s followed by n − ri 0’s. If S is monotone, the discrepancy d(A) of A is the number of posit...

Improved Algorithmic Bounds for Discrepancy of Sparse Set Systems

We consider the problem of finding a low discrepancy coloring for sparse set systems where each element lies in at most t sets. We give an algorithm that finds a coloring with discrepancy O((t log n log s)) where s is the maximum cardinality of a set. This improves upon the previous constructive bound of O(t logn) based on algorithmic variants of the partial coloring method, and for small s (e....

On the discrepancy of strongly unimodular matrices

A (0, 1) matrix A is strongly unimodular if A is totally unimodular and every matrix obtained from A by setting a nonzero entry to 0 is also totally unimodular. Here we consider the linear discrepancy of strongly unimodular matrices. It was proved by Lováz, et.al. [5] that for any matrix A, lindisc(A) ≤ herdisc(A). (1) When A is the incidence matrix of a set-system, a stronger inequality holds:...

Linear Discrepancy of Totally Unimodular Matrices

Let p ∈ [1,∞[ and cp = maxa∈[0,1]((1 − a)ap + a(1 − a)p)1/p. We prove that the known upper bound lindiscp(A) ≤ cp for the Lp linear discrepancy of a totally unimodular matrix A is asymptotically sharp, i.e., sup A lindiscp(A) = cp. We estimate cp = p p+1 ( 1 p+1 )1/p (1+εp) for some εp ∈ [0, 2−p+2], hence cp = 1− ln p p (1+ o(1)). We also show that an improvement for smaller matrices as in the ...