The framework of Pearl's Causal Hierarchy (PCH) formalizes three types of reasoning: probabilistic (i.e. purely observational), interventional, and counterfactual, that reflect the progressive sophistication of human thought regarding causation. We investigate the computational complexity aspects of reasoning in this framework focusing mainly on satisfiability problems expressed in probabilistic and causal languages across the PCH. That is, given a system of formulas in the standard probabilistic and causal languages, does there exist a model satisfying the formulas? Our main contribution is to prove the exact computational complexities showing that languages allowing addition and marginalization (via the summation operator) yield NP^{PP}-, PSPACE-, and NEXP-complete satisfiability problems, depending on the level of the PCH. These are the first results to demonstrate a strictly increasing complexity across the PCH: from probabilistic to causal and counterfactual reasoning. On the other hand, in the case of full languages, i.e.~allowing addition, marginalization, and multiplication, we show that the satisfiability for the counterfactual level remains the same as for the probabilistic and causal levels, solving an open problem in the field.

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(n³(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 n³. 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.

Learning the unknown causal parameters of a linear structural causal model is a fundamental task in causal analysis. The task, known as the problem of identification, asks to estimate the parameters of the model from a combination of assumptions on the graphical structure of the model and observational data, represented as a non-causal covariance matrix. In this paper, we give a new sound and complete algorithm for generic identification which runs in polynomial space. By a standard simulation result, namely $\mathsf{PSPACE} \subseteq \mathsf{EXP}$, this algorithm has exponential running time which vastly improves the state-of-the-art double exponential time method using a Gröbner basis approach. The paper also presents evidence that parameter identification is computationally hard in general. In particular, we prove, that the task asking whether, for a given feasible correlation matrix, there are exactly one or two or more parameter sets explaining the observed matrix, is hard for $\forall \mathbb{R}$, the co-class of the existential theory of the reals. In particular, this problem is $\mathsf{coNP}$-hard. To our best knowledge, this is the first hardness result for some notion of identifiability.

We study formal languages which are capable of fully expressing quantitative probabilistic reasoning and do-calculus reasoning for causal effects, from a computational complexity perspective. We focus on satisfiability problems whose instance formulas allow expressing many tasks in probabilistic and causal inference. The main contribution of this work is establishing the exact computational complexity of these satisfiability problems. We introduce a new natural complexity class, named succ∃R, which can be viewed as a succinct variant of the well-studied class ∃R, and show that these problems are complete for succ∃R. Our results imply even stronger limitations on the use of algorithmic methods for reasoning about probabilities and causality than previous state-of-the-art results that rely only on the NP- or ∃R-completeness of the satisfiability problems for some restricted languages.

Linear structural equation models represent direct causal effects as directed edges and confounding factors as bidirected edges. An open problem is to identify the causal parameters from correlations between the nodes. We investigate models, whose directed component forms a tree, and show that there, besides classical instrumental variables, missing cycles of bidirected edges can be used to identify the model. They can yield systems of quadratic equations that we explicitly solve to obtain one or two solutions for the causal parameters of adjacent directed edges. We show how multiple missing cycles can be combined to obtain a unique solution. This results in an algorithm that can identify instances that previously required approaches based on Gröbner bases, which have doubly-exponential time complexity in the number of structural parameters.

In practice the vast majority of causal effect estimations from observational data are computed using adjustment sets which avoid confounding by adjusting for appropriate covariates. Recently several graphical criteria for selecting adjustment sets have been proposed. They handle causal directed acyclic graphs (DAGs) as well as more general types of graphs that represent Markov equivalence classes of DAGs, including completed partially directed acyclic graphs (CPDAGs). Though expressed in graphical language, it is not obvious how the criteria can be used to obtain effective algorithms for finding adjustment sets. In this paper we provide a new criterion which leads to an efficient algorithmic framework to find, test and enumerate covariate adjustments for chain graphs - mixed graphs representing in a compact way a broad range of Markov equivalence classes of DAGs.

Instrumental Variables are a popular way to identify the direct causal effect of a random variable X on a variable Y. Often no single instrumental variable exists, although it is still possible to find a set of generalized instrumental variables (GIVs) and identify the causal effect of all these variables at once. Till now it was not known how to find GIVs systematically or even test efficiently, if given variables satisfy GIV conditions. We provide fast algorithms for searching and testing restricted cases of GIVs. However, we prove that in the most general case it is NP-hard to verify if given variables fulfill the conditions of a general instrumental sets.

In this paper we propose an approach for global vehicle localization that combines visual odometry with map information from OpenStreetMaps to provide robust and accurate estimates for the vehicle's position. The main contribution of this work comes from the incorporation of the map data as an additional cue into the observation model of a Monte Carlo Localization framework. The resulting approach is able to compensate for the drift that visual odometry accumulates over time, significantly improving localization quality. As our results indicate, the proposed approach outperforms current state-of-the-art visual odometry approaches, indicating in parallel the potential that map data can bring to the global localization task.

During the analysis of packet log files from network experiments, the question arises which received packet belongs to which of the potentially many binary identical send events. We discuss this send-receive correlation problem for networks with local broadcast media. We can prove that assigning send and receive events is an NP-complete problem. However, there is a solution algorithm that is exponential only in the number of nodes; if the number of network nodes is fixed, its complexity is polynomial.