Spencer's theorem in nearly input-sparsity time

A celebrated theorem of Spencer states that for every set system S1, . . . , Sm ⊆ [n], there is a coloring of the ground set with ±1 with discrepancy O( n log(m/n + 2)). We provide an algorithm to find such a coloring in near input-sparsity time O(n + m i=1 |Si|). A key ingredient in our work, which may be of independent interest, is a novel width reduction technique for solving linear programs, not of covering/packing type, in near input-sparsity time using the multiplicative weights update method.