A note on the complexity of the causal ordering problem

摘要

In this note we provide a concise report on the complexity of the causalordering problem, originally introduced by Simon to reason about causaldependencies implicit in systems of mathematical equations. We show thatSimon's classical algorithm to infer causal ordering is NP-Hard---anintractability previously guessed but never proven. We present then a detailedaccount based on Nayak's suggested algorithmic solution (the best available),which is dominated by computing transitive closure---bounded in time by$O(|\mathcal V|\cdot |\mathcal S|)$, where $\mathcal S(\mathcal E, \mathcal V)$is the input system structure composed of a set $\mathcal E$ of equations overa set $\mathcal V$ of variables with number of variable appearances (density)$|\mathcal S|$. We also comment on the potential of causal ordering foremerging applications in large-scale hypothesis management and analytics.