A well-studied challenge that arises in the structure learning problem of causal directed acyclic graphs (DAG) is that using observational data, one can only learn the graph up to a "Markov equivalence class" (MEC). The remaining undirected edges have to be oriented using interventions, which can be very expensive to perform in applications. Thus, the problem of minimizing the number of interventions needed to fully orient the MEC has received a lot of recent attention, and is also the focus of this work. Our first result is a new universal lower bound on the number of single-node interventions that any algorithm (whether active or passive) would need to perform in order to orient a given MEC. Our second result shows that this bound is, in fact, within a factor of two of the size of the smallest set of single-node interventions that can orient the MEC. Our lower bound is provably better than previously known lower bounds. Further, using simulations on synthetic graphs and by giving examples of special graph families, we show that our bound is often significantly better. To prove our lower bound, we develop the notion of clique-block shared-parents (CBSP) orderings, which are topological orderings of DAGs without v-structures and satisfy certain special properties. We also use the techniques developed here to extend our results to the setting of multi-node interventions.