Constructive Discrepancy Minimization with Hereditary L2 Guarantees
In discrepancy minimization problems, we are given a family of sets $\mathcal{S} = \{S_1,\dots,S_m\}$, with each $S_i \in \mathcal{S}$ a subset of some universe $U = \{u_1,\dots,u_n\}$ of $n$ elements. The goal is to find a coloring $\chi : U \to \{-1,+1\}$ of the elements of $U$ such that each set $S \in \mathcal{S}$ is colored as evenly as possible. Two classic measures of discrepancy are $\ell_\infty$-discrepancy defined as $\textrm{disc}_\infty(\mathcal{S},\chi):=\max_{S \in \mathcal{S}} | \sum_{u_i \in S} \chi(u_i) |$ and $\ell_2$-discrepancy defined as $\textrm{disc}_2(\mathcal{S},\chi):=\sqrt{(1/|\mathcal{S}|)\sum_{S \in \mathcal{S}} \left(\sum_{u_i \in S}\chi(u_i)\right)^2}$. Breakthrough work by Bansal gave a polynomial time algorithm, based on rounding an SDP, for finding a coloring $\chi$ such that $\textrm{disc}_\infty(\mathcal{S},\chi) = O(\lg n \cdot \textrm{herdisc}_\infty(\mathcal{S}))$ where $\textrm{herdisc}_\infty(\mathcal{S})$ is the hereditary $\ell_\infty$-discrepancy of $\mathcal{S}$. We complement his work by giving a simple $O((m+n)n^2)$ time algorithm for finding a coloring $\chi$ such $\textrm{disc}_2(\mathcal{S},\chi) = O(\sqrt{\lg n} \cdot \textrm{herdisc}_2(\mathcal{S}))$ where $\textrm{herdisc}_2(\mathcal{S})$ is the hereditary $\ell_2$-discrepancy of $\mathcal{S}$. Interestingly, our algorithm avoids solving an SDP and instead relies on computing eigendecompositions of matrices. Moreover, we use our ideas to speed up the Edge-Walk algorithm by Lovett and Meka [SICOMP'15]. To prove that our algorithm has the claimed guarantees, we show new inequalities relating $\textrm{herdisc}_\infty$ and $\textrm{herdisc}_2$ to the eigenvalues of the matrix corresponding to $\mathcal{S}$. Our inequalities improve over previous work by Chazelle and Lvov, and by Matousek et al. Finally, we also implement our algorithm and show that it far outperforms random sampling.