The Minimization of Random Hypergraphs

TL;DR

This paper studies the phase transition in the size of minimal edges in a maximum-entropy model of random hypergraphs, revealing structural properties with algorithmic and complexity implications.

Contribution

It introduces a precise analysis of the minimization size in random hypergraphs, including a new tight bound on the Chernoff--Hoeffding inequality for binomial variables.

Findings

Expected minimization size exhibits a phase transition at a specific edge count.

Maximum expected minimization size is on the order of ((1+p)^n/rac{n})

Structural insights inform algorithms for hypergraph minimization and complexity problems.

Abstract

We investigate the maximum-entropy model $\mathcal{B}_{n,m,p}$ for random $n$-vertex, $m$-edge multi-hypergraphs with expected edge size $pn$. We show that the expected size of the minimization of $\mathcal{B}_{n,m,p}$, i.e., the number of its inclusion-wise minimal edges, undergoes a phase transition with respect to $m$. If $m$ is at most $1/(1-p)^{(1-p)n}$, then the minimization is of size $\Theta(m)$. Beyond that point, for $\alpha$ such that $m = 1/(1-p)^{\alpha n}$ and $\mathrm{H}$ being the entropy function, it is $\Theta(1) \cdot \min\!\left(1, \, \frac{1}{(\alpha\,{-}\,(1-p)) \sqrt{(1\,{-}\,\alpha) n}}\right) \cdot 2^{(\mathrm{H}(\alpha) + (1-\alpha) \log_2 p) n}.$ This implies that the maximum expected size over all $m$ is $\Theta((1+p)^n/\sqrt{n})$. Our structural findings have algorithmic implications for minimizing an input hypergraph, which in turn has applications in the profiling of relational databases as well as for the Orthogonal Vectors problem studied in fine-grained complexity. The main technical tool is an improvement of the Chernoff--Hoeffding inequality, which we make tight up to constant factors. We show that for a binomial variable $X \sim \mathrm{Bin}(n,p)$ and real number $0 < x \le p$, it holds that $\mathrm{P}[X \le xn] = \Theta(1) \cdot \min\!\left(1, \, \frac{1}{(p-x) \sqrt{xn}}\right) \cdot 2^{-\!\mathrm{D}(x \,{\|}\, p) n}$, where $\mathrm{D}$ denotes the Kullback--Leibler divergence between Bernoulli distributions. The result remains true if $x$ depends on $n$ as long as it is bounded away from $0$.