Spencer's theorem in nearly input-sparsity time
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A celebrated theorem of Spencer states that for every set system S_1,…, S_m ⊆ [n], there is a coloring of the ground set with {± 1} with discrepancy O(√(nlog(m/n+2))). We provide an algorithm to find such a coloring in near input-sparsity time Õ(n+∑_i=1^m|S_i|). 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.
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