30 Oct 2020
•
Bläsius Thomas
•
Friedrich Tobias
•
Schirneck Martin

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 $0$.(read more)