AI Networking Cookbook: Practical recipes for AI-assisted network automation and development
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A shift-chain is an $ r $-uniform hypergraph $ \mathcal{H} $ on vertex set $ [n] $ with the property that, for any two edges $ \{ e_1, \ldots, e_r \} $ and $ \{ f_1, \ldots, f_r \} $ with $ e_1<\cdots<e_r $ and $ f_1<\cdots<f_r $, either $ e_i \le f_i $ for all $ i \in [r] $ or $ f_i \le e_i $ for all $ i \in [r] $. It...
We prove that, whenever $ p \ge n^{-1/2 + o(1)} $, with high probability $ G(n, p) $ admits a fractional triangle decomposition, that is, a non-negative weight function on its triangles for which the total weight of all triangles containing each edge is equal to 1. This bound on $ p $ is optimal up to the asymptotic er...
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