Polytree

Abstract: We consider the task of learning the maximum-likelihood polytree from data. Our first result is a performance guarantee establishing that the optimal branching (or Chow-Liu tree), which can be computed very easily, constitutes a good approximation to the best polytree. We then show that it is not possible to do very much better, since the learning problem is NP-hard even to approximately solve within some constant factor.

Abstract: Many real-world decision making tasks require us to choose among several expensive observations. In a sensor network, for example, it is important to select the subset of sensors that is expected to provide the strongest reduction in uncertainty. In medical decision making tasks, one needs to select which tests to administer before deciding on the most effective treatment. It has been general practice to use heuristic-guided procedures for selecting observations. In this paper, we present the first efficient optimal algorithms for selecting observations for a class of probabilistic graphical models. For example, our algorithms allow to optimally label hidden variables in Hidden Markov Models (HMMs). We provide results for both selecting the optimal subset of observations, and for obtaining an optimal conditional observation plan. Furthermore we prove a surprising result: In most graphical models tasks, if one designs an efficient algorithm for chain graphs, such as HMMs, this procedure can be generalized to polytree graphical models. We prove that the optimizing value of information is $NP^{PP}$-hard even for polytrees. It also follows from our results that just computing decision theoretic value of information objective functions, which are commonly used in practice, is a #P-complete problem even on Naive Bayes models (a simple special case of polytrees). In addition, we consider several extensions, such as using our algorithms for scheduling observation selection for multiple sensors. We demonstrate the effectiveness of our approach on several real-world datasets, including a prototype sensor network deployment for energy conservation in buildings.

Abstract: Unary operator domains - i.e., domains in which operators have a single effect - arise naturally in many control problems. In its most general form, the problem of STRIPS planning in unary operator domains is known to be as hard as the general STRIPS planning problem - both are PSPACE-complete. However, unary operator domains induce a natural structure, called the domain's causal graph. This graph relates between the preconditions and effect of each domain operator. Causal graphs were exploited by Williams and Nayak in order to analyze plan generation for one of the controllers in NASA's Deep-Space One spacecraft. There, they utilized the fact that when this graph is a tree, a serialization ordering over any subgoal can be obtained quickly. In this paper we conduct a comprehensive study of the relationship between the structure of a domain's causal graph and the complexity of planning in this domain. On the positive side, we show that a non-trivial polynomial time plan generation algorithm exists for domains whose causal graph induces a polytree with a constant bound on its node indegree. On the negative side, we show that even plan existence is hard when the graph is a singly connected DAG. More generally, we show that the number of paths in the causal graph is closely related to the complexity of planning in the associated domain. Finally we relate our results to the question of complexity of planning with serializable subgoals.