A spectral bound on hypergraph discrepancy

07/07/2019

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by   Aditya Potukuchi, et al.

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Let H be a t-regular hypergraph on n vertices and m edges. Let M be the m × n incidence matrix of H and let us denote λ =_v ∈1^1/√(t)vMv. We show that the discrepancy of H is O(√(λ t)). As a corollary, this gives us that for every t, the discrepancy of a random t-regular hypergraph with n vertices and m ≥ n edges is almost surely O(√(t)) as n grows. The proof also gives a polynomial time algorithm that takes a hypergraph as input and outputs a coloring with the above guarantee.