  Counting and sampling directed acyclic graphs from a Markov equivalence class
are fundamental tasks in graphical causal analysis. In this paper we show that
these tasks can be performed in polynomial time, solving a long-standing open
problem in this area. Our algorithms are effective and easily implementable. As
we show in experiments, these breakthroughs make thought-to-be-infeasible
strategies in active learning of causal structures and causal effect
identification with regard to a Markov equivalence class practically
applicable.
