Causal effect estimation from observational data is a fundamental task in empirical sciences. It becomes particularly challenging when unobserved confounders are involved in a system. This paper focuses on front-door adjustment -- a classic technique which, using observed mediators allows to identify causal effects even in the presence of unobserved confounding. While the statistical properties of the front-door estimation are quite well understood, its algorithmic aspects remained unexplored for a long time. In 2022, Jeong, Tian, and Bareinboim presented the first polynomial-time algorithm for finding sets satisfying the front-door criterion in a given directed acyclic graph (DAG), with an $O(n^3(n+m))$ run time, where $n$ denotes the number of variables and $m$ the number of edges of the causal graph. In our work, we give the first linear-time, i.e., $O(n+m)$, algorithm for this task, which thus reaches the asymptotically optimal time complexity. This result implies an $O(n(n+m))$ delay enumeration algorithm of all front-door adjustment sets, again improving previous work by a factor of $n^3$. Moreover, we provide the first linear-time algorithm for finding a minimal front-door adjustment set. We offer implementations of our algorithms in multiple programming languages to facilitate practical usage and empirically validate their feasibility, even for large graphs.

翻译：从观测数据中估计因果效应是实证科学中的基本任务。当系统中存在未观测混杂因素时，这一任务尤其具有挑战性。本文聚焦于前门调整——这一经典技术通过使用观测中介变量，即使在存在未观测混杂的情况下也能识别因果效应。尽管前门估计的统计性质已得到较好理解，但其算法层面长期未被探索。2022年，Jeong、Tian和Bareinboim提出了首个多项式时间算法，用于在给定有向无环图（DAG）中寻找满足前门准则的集合，其运行时间为$O(n^3(n+m))$，其中$n$表示变量数量，$m$表示因果图的边数。在本研究中，我们针对该任务提出了首个线性时间算法（即$O(n+m)$），从而达到了渐近最优时间复杂度。这一结果进一步导出了所有前门调整集合的$O(n(n+m))$延迟枚举算法，再次将先前工作的复杂度降低$n^3$倍。此外，我们还提供了首个用于寻找最小前门调整集合的线性时间算法。我们以多种编程语言实现了所提出的算法，以促进实际应用，并通过实验验证了这些算法在大规模图上的可行性。