Informed Greedy Algorithm for Scalable Bayesian Network Fusion via Minimum Cut Analysis

This paper presents the Greedy Min-Cut Bayesian Consensus (GMCBC) algorithm for the structural fusion of Bayesian Networks (BNs). The method is designed to preserve essential dependencies while controlling network complexity. It addresses the limitations of traditional fusion approaches, which often lead to excessively complex models that are impractical for inference, reasoning, or real-world applications. As the number and size of input networks increase, this issue becomes even more pronounced. GMCBC integrates principles from flow network theory into BN fusion, adapting the Backward Equivalence Search (BES) phase of the Greedy Equivalence Search (GES) algorithm and applying the Ford-Fulkerson algorithm for minimum cut analysis. This approach removes non-essential edges, ensuring that the fused network retains key dependencies while minimizing unnecessary complexity. Experimental results on synthetic Bayesian Networks demonstrate that GMCBC achieves near-optimal network structures. In federated learning simulations, GMCBC produces a consensus network that improves structural accuracy and dependency preservation compared to the average of the input networks, resulting in a structure that better captures the real underlying (in)dependence relationships. This consensus network also maintains a similar size to the original networks, unlike unrestricted fusion methods, where network size grows exponentially.

A consensus set for the aggregation of partial rankings: the case of the Optimal Set of Bucket Orders Problem

In rank aggregation problems (RAP), the solution is usually a consensus ranking that generalizes a set of input orderings. There are different variants that differ not only in terms of the type of rankings that are used as input and output, but also in terms of the objective function employed to evaluate the quality of the desired output ranking. In contrast, in some machine learning tasks (e.g. subgroup discovery) or multimodal optimization tasks, attention is devoted to obtaining several models/results to account for the diversity in the input data or across the search landscape. Thus, in this paper we propose to provide, as the solution to an RAP, a set of rankings to better explain the preferences expressed in the input orderings. We exemplify our proposal through the Optimal Bucket Order Problem (OBOP), an RAP which consists in finding a single consensus ranking (with ties) that generalizes a set of input rankings codified as a precedence matrix. To address this, we introduce the Optimal Set of Bucket Orders Problem (OSBOP), a generalization of the OBOP that aims to produce not a single ranking as output but a set of consensus rankings. Experimental results are presented to illustrate this proposal, showing how, by providing a set of consensus rankings, the fitness of the solution significantly improves with respect to the one of the original OBOP, without losing comprehensibility.