Induced acyclic tournaments in random digraphs: Sharp concentration, thresholds and algorithms

Given a simple directed graph $D = (V,A)$, let the size of the largest induced acyclic tournament be denoted by $mat(D)$. Let $D ∈ \mathcal{D}(n, p)$ (with $p = p(n)$) be a random instance, obtained by randomly orienting each edge of a random graph drawn from $\mathcal{G}(n, 2p)$. We show that $mat(D)$ is asymptotically almost surely (a.a.s.) one of only 2 possible values, namely either $b^\ast$ or $b^\ast + 1$, where $b^\ast = ⌊2(log_rn) + 0.5⌋$ and $r = p^{−1}$. It is also shown that if, asymptotically, $2(log_rn) + 1$ is not within a distance of $w(n)//(ln n)$ (for any sufficiently slow $w(n) → ∞$) from an integer, then $mat(D)$ is $⌊2(log_rn) + 1⌋$ a.a.s. As a consequence, it is shown that $mat(D)$ is 1-point concentrated for all $n$ belonging to a subset of positive integers of density 1 if $p$ is independent of $n$. It is also shown that there are functions $p = p(n)$ for which $mat(D)$ is provably not concentrated in a single value. We also establish thresholds (on $p$) for the existence of induced acyclic tournaments of size i which are sharp for $i = i(n) → ∞$. We also analyze a polynomial time heuristic and show that it produces a solution whose size is at least $log_rn + Θ(\sqrt{log_rn})$. Our results are valid as long as $p ≥ 1//n$. All of these results also carry over (with some slight changes) to a related model which allows 2-cycles.