A Bayesian network is a directed acyclic graph that represents statistical dependencies between variables of a joint probability distribution. A fundamental task in data science is to learn a Bayesian network from observed data. Polytree Learning is the problem of learning an optimal Bayesian network that fulfills the additional property that its underlying undirected graph is a forest. In this work, we revisit the complexity of Polytree Learning. We show that Polytree Learning can be solved in single-exponential FPT time for the number of variables. Moreover, we consider the influence of d, the number of variables that might receive a nonempty parent set in the final DAG on the complexity of Polytree Learning. We show that Polytree Learning is presumably not fixed-parameter tractable for d, unlike Bayesian network learning which is fixed-parameter tractable for d. Finally, we show that if d and the maximum parent set size are bounded, then we can obtain efficient algorithms.
We introduce and study WEIGHTED MIN (s, t)-CUT PREVENTION, where we are given a graph G = (V, E) with vertices s and t and an edge cost function and the aim is to choose an edge set D of total cost at most d such that G has no (s, t)-edge cut of capacity at most a that is disjoint from D. We show that WEIGHTED MIN (s, t)-CUT PREVENTION is NP-hard even on subcubcic graphs when all edges have capacity and cost one and provide a comprehensive study of the parameterized complexity of the problem. We show, for example W[1]-hardness with respect to d and an FPT algorithm for a.
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