On the discrepancy of set systems definable in sparse graph classes

Discrepancy is a natural measure for the inherent complexity of set systems with many applications in mathematics and computer science. The discrepancy of a set system $(U,\mathscr S)$ is the minimum over all mappings $\chi\colon U\rightarrow\{-1,1\}$ of $\max_{S\in\mathscr S}\bigl|\sum_{v\in S}\chi(v)\bigr|$. We study the discrepancy of set systems that are first-order definable in sparse graph classes. We prove that all the set systems definable in a monotone class $\mathscr C$ have bounded discrepancy if and only if $\mathscr C$ has bounded expansion, and that they have hereditary discrepancy at most $|U|^{c}$ (for some~$c<1/2$) if and only if $\mathscr C$ is nowhere dense. However, if $\mathscr C$ is somewhere dense, then for every positive integer $d$ there is a set system of $d$-tuples definable in $\mathscr C$ with discrepancy $\Omega(|U|^{1/2})$. From the algorithmic point of view, we prove that if $\mathscr C$ is a class of graphs with bounded expansion and $\phi(\bar x;\bar y)$ is a first-order formula, then for each input graph $G\in\mathscr C$, a mapping $\chi:V(G)^{|\bar x|}\rightarrow\{-1,1\}$ witnessing the boundedness of the discrepancy of the set-system defined by~$\phi$ can be computed in $\mathcal O(|G|^{|\bar x|})$ time. We also deduce that for such set-systems, when $|\bar x|=1$, $\varepsilon$-nets of size $\mathcal{O}(1/\varepsilon)$ can be computed in time $\mathcal{O}(|G|\,\log |G|)$ and $\varepsilon$-approximations of size $\mathcal{O}(1/\varepsilon)$ can be computed in polynomial time.