Largest Induced Acyclic Tournament in Random Digraphs: A 2-Point Concentration

Given a simple directed graph D=(V,A), let the size of the largest induced acyclic tournament be denoted by mat(D). Let $D \in \mathcal{D}(n,p)$ be a random instance, obtained by choosing each of the ${{n}\choose{2}}$ possible undirected edges independently with probability 2p and then orienting each chosen edge in one of two possible directions with probability 1/2. We show that for such a random instance, mat(D) is asymptotically almost surely one of only 2 possible values, namely either b* or b*+1, where $b^* = \lfloor 2(\log_{p^{-1}} n)+0.5 \rfloor$. It is then shown that almost surely any maximal induced acyclic tournament is of a size which is at least nearly half of any optimal solution. We also analyze a polynomial time heuristic and show that almost surely it produces a solution whose size is at least $\log_{p^{-1}} n + \Theta(\sqrt{\log_{p^{-1}} n})$. Our results also carry over to a related model in which each possible directed arc is chosen independently with probability p. An immediate corollary is that (the size of a) minimum feedback vertex set can be approximated within a ratio of 1+o(1) for random tournaments.