Structure Approximation of Most Probable Explanations in Bayesian Networks

Abstract Typically, when one discusses approximation algorithms for (NP-hard) problems (like TRAVELING SALESPERSON, VERTEX COVER, KNAPSACK), one refers to algorithms that return a solution whose value is (at least ideally) close to optimal; e.g., a tour with almost minimal length, a vertex cover of size just above minimal, or collection of objects that has close to maximal value. In contrast, one might also be interested in approximations algorithms that return solutions that resemble the optimal solutions, i.e., whose structure is akin to the optimal solution, like a tour that is almost similar to the optimal tour, a vertex cover that differs in only a few vertices from the optimal cover, or a collection that is similar to the optimal collection. In this paper, we discuss structure-approximation of the problem of finding the most probable explanation of observations in Bayesian networks, i.e., finding a joint value assignment that looks like the most probable one, rather than has an almost as high value. We show that it is NP-hard to obtain the value of just a single variable of the most probable explanation. However, when partial orders on the values of the variables are available, we can improve on these results.