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تسجيل مستخدم جديدA spectral bound on hypergraph discrepancy
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Aditya Potukuchi
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Let $mathcal{H}$ be a $t$-regular hypergraph on $n$ vertices and $m$ edges. Let $M$ be the $m times n$ incidence matrix of $mathcal{H}$ and let us denote $lambda =max_{v perp overline{1},|v| = 1}|Mv|$. We show that the discrepancy of $mathcal{H}$ is $O(sqrt{t} + lambda)$. As a corollary, this gives us that for every $t$, the discrepancy of a random $t$-regular hypergraph with $n$ vertices and $m geq n$ edges is almost surely $O(sqrt{t})$ 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.
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