Given a simple directed graph $D = (V,A)$, let the size of the largest induced directed acyclic graph $\textit{(dag)}$ be denoted by $mas(D)$. Let $D \in \mathcal{D}(n,p)$ be a $\textit{random}$ instance, obtained by choosing each of the $\binom{n}{2}$ possible undirected edges independently with probability $2p$ and then orienting each chosen edge independently in one of two possible directions with probabibility $1/2$. We obtain improved bounds on the range of concentration, upper and lower bounds of $mas(D)$. Our main result is that $mas(D) \geq \lfloor 2\log_q np - X \rfloor$ where $q = (1-p)^{-1}, X=W$ if $p \geq n^{-1/3+\epsilon}$ ($\epsilon > 0$ is any constant), $X=W/(\ln q)$ if $p \geq n^{-1/2}(\ln n)^2$, and $W$ is a suitably large constant. where we have an $O(\ln \ln np/\ln q)$ term instead of $W$. This improves the previously known lower bound with an $O(\ln \ln np/\ln q)$ term instead of $W$. We also obtain a slight improvement on the upper bound, using an upper bound on the number of acyclic orientations of an undirected graph. We also analyze a polynomial-time heuristic to find a large induced dag and show that it produces a solution whose size is at least $\log _q np + \Theta (\sqrt{\log_q np})$.
Discrete Mathematics & Theoretical Computer Science