Deterministic Discrepancy Minimization

We derandomize a recent algorithmic approach due to Bansal (Foundations of Computer Science, FOCS, pp. 3–10, 2010) to efficiently compute low discrepancy colorings for several problems, for which only existential results were previously known. In particular, we give an efficient deterministic algorithm for Spencer’s six standard deviations result (Spencer in Trans. Am. Math. Soc. 289:679–706, 1985), and to find a low discrepancy coloring for a set system with low hereditary discrepancy. The main new idea is to add certain extra constraints to the natural semidefinite programming formulation for discrepancy, which allow us to argue about the existence of a good deterministic move at each step of the algorithm. The non-constructive entropy method is used to argue the feasibility of this enhanced SDP.