The Minimization of Random Hypergraphs

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 $\min(\mathcal{B}_{n,m,p})$, i.e., the number of inclusion-wise minimal edges of $\mathcal{B}_{n,m,p}$, undergoes a phase transition with respect to $m$. If $m$ is at most $1/(1-p)^{(1-p)n}$, then $\mathrm{E}[|\min(\mathcal{B}_{n,m,p})|]$ is of order $\Theta(m)$, while for $m \ge 1/(1-p)^{(1-p+\varepsilon)n}$ for any $\varepsilon > 0$, it is $\Theta( 2^{(\mathrm{H}(\alpha) + (1-\alpha) \log_2 p) n}/ \sqrt{n})$. Here, $\mathrm{H}$ denotes the binary entropy function and $\alpha = - (\log_{1-p} m)/n$. The result implies that the maximum expected number of minimal edges over all $m$ is $\Theta((1+p)^n/\sqrt{n})$. Our structural findings have algorithmic implications for minimizing an input hypergraph, which has applications in the profiling of relational databases as well as for the Orthogonal Vectors problem studied in fine-grained complexity. We make several technical contributions that are of independent interest in probability. First, we improve the Chernoff--Hoeffding theorem on the tail of the binomial distribution. In detail, we show that for a binomial variable $Y \sim \operatorname{Bin}(n,p)$ and any $0 < x < p$, it holds that $\mathrm{P}[Y \le xn] = \Theta( 2^{-\!\mathrm{D}(x \,{\|}\, p) n}/\sqrt{n})$, where $\mathrm{D}$ is the binary Kullback--Leibler divergence between Bernoulli distributions. We give explicit upper and lower bounds on the constants hidden in the big-O notation that hold for all $n$. Secondly, we establish the fact that the probability of a set of cardinality $i$ being minimal after $m$ i.i.d. maximum-entropy trials exhibits a sharp threshold behavior at $i^* = n + \log_{1-p} m$.