A size-sensitive discrepancy bound for set systems of bounded primal shatter dimension

Let (X, δ) be a set system on an n-point set X. The discrepancy of δ is defined as the minimum of the largest deviation from an even split, over all subsets of δ ∈ δ and two-colorings χ on X. We consider the scenario where, for any subset X' ⊆ X of size m ≤ n and for any parameter 1 ≤ k ≤ m, the number of restrictions of the sets of δ to X' of size at most k is only 0(m^d1k^d-d1), for fixed integers d > 0 and 1 ≤ d₁ ≤ d (this generalizes the standard notion of bounded primal shatter dimension when d₁ = d). In this case we show that there exists a coloring χ with discrepancy bound o^*(|S|^1/2-d₁/(2d)_n(d ₁-1)(2d), for each S ∈ δ, where O ^*(·) hides a polylogarithmic factor in n. This bound is tight up to a polylogarithmic factor [21, 23] and the corresponding coloring χ can be computed in expected polynomial time using the very recent machinery of Lovett and Meka for constructive discrepancy minimization [20]. Our bound improves and generalizes the bounds obtained from the machinery of Har-Peled and Sharir [15] (and the follow-up work in [27]) for points and halfspaces in d- space for d ≥ 3. Last but not least, we show that our bound yields improved bounds for the size of relative (ε, δ)- Approximations for set systems of the above kind.