Constructive Algorithms for Discrepancy Minimization

Abstract

Given a set system $(V,\mathcal{S})$, $V=\{1,\ldots,n\}$ and $\mathcal{S}=\{S_1,\ldots,S_m\}$, the minimum discrepancy problem is to find a 2-coloring $\mathcal{X}:V \right arrow \{-1,+1\}$, such that each set is colored as evenly as possible, i.e. find $\mathcal{X}$ to minimize $\max_{j \in [m]} \left|\sum_{i \in S_j} \mathcal{X}(i)\right|$. 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. Specifically we give efficient randomized algorithms to: 1. Construct an $O(n^{1/2})$ discrepancy coloring for general sets systems when $m=O(n)$, matching the celebrated result of Spencer up to $O(1)$ factors. More generally, for $m\geq n$, we obtain a discrepancy of $O(n^{1/2} \log (2m/n))$. 2. Construct a coloring with discrepancy $O(t^{1/2} \log n)$, if each element lies in at most $t$ sets. This matches the (non-constructive) result of Srinivasan. 3. Construct a coloring with discrepancy $O( \lambda\log (nm))$, where $\lambda$ is the hereditary discrepancy of the set system. 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 $\pm 1$. At each step the random hops for various elements are correlated by a solution to a semi definite program, where this program is determined by the current state and the entropy method.