Constructive Algorithms for Discrepancy Minimization

Given a set system (V,S), V={1,...,n} and S={S1,...,Sm}, the minimum discrepancy problem is to find a 2-coloring of V, such that each set is colored as evenly as possible. In this paper we give the first polynomial time algorithms for discrepancy minimization that achieve bounds similar to those known existentially using the so-called Entropy Method. We also give a first approximation-like result for discrepancy. The main idea in our algorithms is to produce a coloring over time by letting the color of the elements perform a random walk (with tiny increments) starting from 0 until they reach $-1$ or $+1$. At each time step the random hops for various elements are correlated using the solution to a semidefinite program, where this program is determined by the current state and the entropy method.