On the Linear and Hereditary Discrepancies

We exhibit a set system S 0 such that any set system containing it has linear discrepancy at least 2. Let V be a nite set and S 2 V a system of subsets of V. The incidence matrix A of (V; S) has rows indexed by the sets of S and the columns indexed by the points of V. The column corresponding to a point v 2 V is denoted by a v. A coloring of V is a mapping : V ! f?1; +1g. The discrepancy of S, denoted by disc(S), is given by disc(S) = min disc(; S); where the minimum is over all colorings of V and disc(; S) = max S2S j(S)j, with (S) = P v2S (v). The hereditary discrepancy of S, denoted by herdisc(S), is herdisc(S) = max UV disc(Sj U); where Sj U = fS \ U : S 2 Sg. (In the sequel, by saying that a set system S contains a set system S 0 we mean that S 0 Sj U for some subset U of the ground set of S.) The linear discrepancy arises in the following \rounding" problem. Each point v 2 V is assigned a weight w v 2 ?1; 1]. We want a coloring of V for which the sum of the colors in each set S 2 S is close to the total weight of its points. The discrepancy of S with respect to the given weights is thus min :V !f?1;1g max S2S j(S) ? w(S)j ; and the linear discrepancy of S is the supremum of this quantity over all choices of the weight vector w 2 ?1; 1] V. These deenitions can be expressed in terms of the incidence matrix A of S, and in that form, they can be used to deene the various discrepancies for arbitrary real matrices A. Namely, for an m n matrix A we have disc(A) = min x2f?1;1g n kAxk 1 ; herdisc(A) = max B disc(B) where B ranges over all submatrices of A, and lindisc(A) = max w2?1;1] n min x2f?1;1g n kA(x ? w)k 1 :