Constructive Discrepancy Minimization with Hereditary L2 Guarantees
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In discrepancy minimization problems, we are given a family of sets S = {S_1,...,S_m}, with each S_i ∈S a subset of some universe U = {u_1,...,u_n} of n elements. The goal is to find a coloring χ : U →{-1,+1} of the elements of U such that each set S ∈S is colored as evenly as possible. Two classic measures of discrepancy are ℓ_∞-discrepancy defined as disc_∞(S,χ):=_S ∈S | ∑_u_i ∈ Sχ(u_i) | and ℓ_2-discrepancy defined as disc_2(S,χ):=√((1/|S|)∑_S ∈S(∑_u_i ∈ Sχ(u_i))^2). Breakthrough work by Bansal gave a polynomial time algorithm, based on rounding an SDP, for finding a coloring χ such that disc_∞(S,χ) = O( n ·herdisc_∞(S)) where herdisc_∞(S) is the hereditary ℓ_∞-discrepancy of S. We complement his work by giving a polynomial time algorithm for finding a coloring χ such disc_2(S,χ) = O(√( n)·herdisc_2(S)) where herdisc_2(S) is the hereditary ℓ_2-discrepancy of S. Interestingly, our algorithm avoids solving an SDP and instead relies simply on computing eigendecompositions of matrices. To prove that our algorithm has the claimed guarantees, we also prove new inequalities relating both herdisc_∞ and herdisc_2 to the eigenvalues of the incidence matrix corresponding to S. Our inequalities improve over previous work by Chazelle and Lvov, and by Matousek, Nikolov and Talwar. We believe these inequalities are of independent interest as powerful tools for proving hereditary discrepancy lower bounds.
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