Fast Discrepancy Minimization with Hereditary Guarantees (2207.03268v2)

Abstract: Efficiently computing low discrepancy colorings of various set systems, has been studied extensively since the breakthrough work by Bansal (FOCS 2010), who gave the first polynomial time algorithms for several important settings, including for general set systems, sparse set systems and for set systems with bounded hereditary discrepancy. The hereditary discrepancy of a set system, is the maximum discrepancy over all set systems obtainable by deleting a subset of the ground elements. While being polynomial time, Bansal's algorithms were not practical, with e.g. his algorithm for the hereditary setup running in time $\Omega(m n^{4.5})$ for set systems with $m$ sets over a ground set of $n$ elements. More efficient algorithms have since then been developed for general and sparse set systems, however, for the hereditary case, Bansal's algorithm remains state-of-the-art. In this work, we give a significantly faster algorithm with hereditary guarantees, running in $O(mn^2\lg(2 + m/n) + n^3)$ time. Our algorithm is based on new structural insights into set systems with bounded hereditary discrepancy. We also implement our algorithm and show experimentally that it computes colorings that are significantly better than random and finishes in a reasonable amount of time, even on set systems with thousands of sets over a ground set of thousands of elements.