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Bayesian network structure learning is the notoriously difficult problem of discovering a Bayesian network that optimally represents a given set of training data. In this paper we study the computational worst-case complexity of exact Bayesian network structure learning under graph theoretic restrictions on the (directed) super-structure. The super-structure is an undirected graph that contains as subgraphs the skeletons of solution networks. We introduce the directed super-structure as a natural generalization of its undirected counterpart. Our results apply to several variants of score-based Bayesian network structure learning where the score of a network decomposes into local scores of its nodes. Results: We show that exact Bayesian network structure learning can be carried out in non-uniform polynomial time if the super-structure has bounded treewidth, and in linear time if in addition the super-structure has bounded maximum degree. Furthermore, we show that if the directed super-structure is acyclic, then exact Bayesian network structure learning can be carried out in quadratic time. We complement these positive results with a number of hardness results. We show that both restrictions (treewidth and degree) are essential and cannot be dropped without loosing uniform polynomial time tractability (subject to a complexity-theoretic assumption). Similarly, exact Bayesian network structure learning remains NP-hard for “almost acyclic” directed super-structures. Furthermore, we show that the restrictions remain essential if we do not search for a globally optimal network but aim to improve a given network by means of at most k arc additions, arc deletions, or arc reversals (k-neighborhood local search).
Parameterized Complexity Results for Exact Bayesian Network Structure Learning

In many applications spanning from sensor to social networks, transportation systems, gene regulatory networks or big data, the signals of interest are defined over the vertices of a graph. The aim of this paper is to propose a least mean square (LMS) strategy for adaptive estimation of signals defined over graphs. Assuming the graph signal to be band-limited, over a known bandwidth, the method enables reconstruction, with guaranteed performance in terms of mean-square error, and tracking from a limited number of observations over a subset of vertices. A detailed mean square analysis provides the performance of the proposed method, and leads to several insights for designing useful sampling strategies for graph signals. Numerical results validate our theoretical findings, and illustrate the performance of the proposed method. Furthermore, to cope with the case where the bandwidth is not known beforehand, we propose a method that performs a sparse online estimation of the signal support in the (graph) frequency domain, which enables online adaptation of the graph sampling strategy. Finally, we apply the proposed method to build the power spatial density cartography of a given operational region in a cognitive network environment.
Least Mean Squares Estimation of Graph Signals

In the election of a hierarchical clustering method, theoretic properties may give some insight to determine which method is the most suitable to treat a clustering problem. Herein, we study some basic properties of two hierarchical clustering methods: α-unchaining single linkage or SL(α) and a modified version of this one, SL∗(α). We compare the results with the properties satisfied by the classical linkage-based hierarchical clustering methods.

ON THE PROPERTIES OF α-UNCHAINING SINGLE LINKAGE HIERARCHICAL CLUSTERING