A tree with $n$ vertices has at most $95^{n/13}$ minimal dominating sets. The growth constant $λ= \sqrt[13]{95} \approx 1.4194908$ is best possible. It is obtained in a semi-automatic way as a kind of "dominant eigenvalue" of a bilinear operation on sixtuples that is derived from the dynamic-programming recursion for computing the number of minimal dominating sets of a tree. We also derive an output-sensitive algorithm for listing all minimal dominating sets with linear set-up time and linear delay between successive solutions.

Causal inference aids researchers in discovering cause-and-effect relationships, leading to scientific insights. Accurate causal estimation requires identifying confounding variables to avoid false discoveries. Pearl's causal model uses causal DAGs to identify confounding variables, but incorrect DAGs can lead to unreliable causal conclusions. However, for high dimensional data, the causal DAGs are often complex beyond human verifiability. Graph summarization is a logical next step, but current methods for general-purpose graph summarization are inadequate for causal DAG summarization. This paper addresses these challenges by proposing a causal graph summarization objective that balances graph simplification for better understanding while retaining essential causal information for reliable inference. We develop an efficient greedy algorithm and show that summary causal DAGs can be directly used for inference and are more robust to misspecification of assumptions, enhancing robustness for causal inference. Experimenting with six real-life datasets, we compared our algorithm to three existing solutions, showing its effectiveness in handling high-dimensional data and its ability to generate summary DAGs that ensure both reliable causal inference and robustness against misspecifications.

Acyclic schemes posses known benefits for database design, speeding up queries, and reducing space requirements. An acyclic join dependency (AJD) is lossless with respect to a universal relation if joining the projections associated with the schema results in the original universal relation. An intuitive and standard measure of loss entailed by an AJD is the number of redundant tuples generated by the acyclic join. Recent work has shown that the loss of an AJD can also be characterized by an information-theoretic measure. Motivated by the problem of automatically fitting an acyclic schema to a universal relation, we investigate the connection between these two characterizations of loss. We first show that the loss of an AJD is captured using the notion of KL-Divergence. We then show that the KL-divergence can be used to bound the number of redundant tuples. We prove a deterministic lower bound on the percentage of redundant tuples. For an upper bound, we propose a random database model, and establish a high probability bound on the percentage of redundant tuples, which coincides with the lower bound for large databases.

We study the \em generalized model counting problem, defined as follows: given a database, and a set of deterministic tuples, count the number of subsets of the database that include all deterministic tuples and satisfy the query. This problem is computationally equivalent to the evaluation of the query over a tuple-independent probabilistic database where all tuples have probabilities in $\set0,\frac1 2, 1 $. Previous work has established a dichotomy for Unions of Conjunctive Queries (UCQ) when the probabilities are arbitrary rational numbers, showing that, for each query, its complexity is either in polynomial time or \#P-hard. The query is called \em safe in the first case, and \em unsafe in the second case. Here, we strengthen the hardness proof, by proving that an unsafe UCQ query remains \#P-hard even if the probabilities are restricted to $\set0,\frac1 2, 1 $. This requires a complete redesign of the hardness proof, using new techniques. A related problem is the \em model counting problem, which asks for the probability of the query when the input probabilities are restricted to $\set0,\frac1 2 $. While our result does not extend to model counting for all unsafe UCQs, we prove that model counting is \#P-hard for a class of unsafe queries called Type-I forbidden queries.

Integrity constraints such as functional dependencies (FD), and multi-valued dependencies (MVD) are fundamental in database schema design. Likewise, probabilistic conditional independences (CI) are crucial for reasoning about multivariate probability distributions. The implication problem studies whether a set of constraints (antecedents) implies another constraint (consequent), and has been investigated in both the database and the AI literature, under the assumption that all constraints hold exactly. However, many applications today consider constraints that hold only approximately. In this paper we define an approximate implication as a linear inequality between the degree of satisfaction of the antecedents and consequent, and we study the relaxation problem: when does an exact implication relax to an approximate implication? We use information theory to define the degree of satisfaction, and prove several results. First, we show that any implication from a set of data dependencies (MVDs+FDs) can be relaxed to a simple linear inequality with a factor at most quadratic in the number of variables; when the consequent is an FD, the factor can be reduced to 1. Second, we prove that there exists an implication between CIs that does not admit any relaxation; however, we prove that every implication between CIs relaxes "in the limit". Finally, we show that the implication problem for differential constraints in market basket analysis also admits a relaxation with a factor equal to 1. Our results recover, and sometimes extend, several previously known results about the implication problem: implication of MVDs can be checked by considering only 2-tuple relations, and the implication of differential constraints for frequent item sets can be checked by considering only databases containing a single transaction.

Acyclic schemes have numerous applications in databases and in machine learning, such as improved design, more efficient storage, and increased performance for queries and machine learning algorithms. Multivalued dependencies (MVDs) are the building blocks of acyclic schemes. The discovery from data of both MVDs and acyclic schemes is more challenging than other forms of data dependencies, such as Functional Dependencies, because these dependencies do not hold on subsets of data, and because they are very sensitive to noise in the data; for example a single wrong or missing tuple may invalidate the schema. In this paper we present Maimon, a system for discovering approximate acyclic schemes and MVDs from data. We give a principled definition of approximation, by using notions from information theory, then describe the two components of Maimon: mining for approximate MVDs, then reconstructing acyclic schemes from approximate MVDs. We conduct an experimental evaluation of Maimon on 20 real-world datasets, and show that it can scale up to 1M rows, and up to 30 columns.

Models of uncertain preferences, such as Mallows, have been extensively studied due to their plethora of application domains. In a recent work, a conceptual and theoretical framework has been proposed for supporting uncertain preferences as first-class citizens in a relational database. The resulting database is probabilistic, and, consequently, query evaluation entails inference of marginal probabilities of query answers. In this paper, we embark on the challenge of a practical realization of this framework. We first describe an implementation of a query engine that supports querying probabilistic preferences alongside relational data. Our system accommodates preference distributions in the general form of the Repeated Insertion Model (RIM), which generalizes Mallows and other models. We then devise a novel inference algorithm for conjunctive queries over RIM, and show that it significantly outperforms the state of the art in terms of both asymptotic and empirical execution cost. We also develop performance optimizations that are based on sharing computation among different inference tasks in the workload. Finally, we conduct an extensive experimental evaluation and demonstrate that clear performance benefits can be realized by a query engine with built-in probabilistic inference, as compared to a stand alone implementation with a black-box inference solver.

We present an algorithm that enumerates all the minimal triangulations of a graph in incremental polynomial time. Consequently, we get an algorithm for enumerating all the proper tree decompositions, in incremental polynomial time, where ``proper'' means that the tree decomposition cannot be improved by removing or splitting a bag. The algorithm can incorporate any method for (ordinary, single result) triangulation or tree decomposition, and can serve as an anytime algorithm to improve such a method. We describe an extensive experimental study of an implementation on real data from different fields. Our experiments show that the algorithm improves upon central quality measures over the underlying tree decompositions, and is able to produce a large number of high-quality decompositions.

We propose a novel framework wherein probabilistic preferences can be naturally represented and analyzed in a probabilistic relational database. The framework augments the relational schema with a special type of a relation symbol---a preference symbol. A deterministic instance of this symbol holds a collection of binary relations. Abstractly, the probabilistic variant is a probability space over databases of the augmented form (i.e., probabilistic database). Effectively, each instance of a preference symbol can be represented as a collection of parametric preference distributions such as Mallows. We establish positive and negative complexity results for evaluating Conjunctive Queries (CQs) over databases where preferences are represented in the Repeated Insertion Model (RIM), Mallows being a special case. We show how CQ evaluation reduces to a novel inference problem (of independent interest) over RIM, and devise a solver with polynomial data complexity.