Complexity Results for Local Monotonicity in Probabilistic Networks

Often, monotonicity is a desirable property of probabilistic networks. For example, when medical knowledge in a particular domain dictates that more severe symptoms increase the likeliness of a more serious disease, these properties should be reflected in the network. Unfortunately, the problem to determine for a given probabilistic network whether it is monotone is known to be a highly intractable problem. Often, approximation algorithms are employed that work on a local scale. These algorithms determine the monotonicity of the arcs, rather than the network as a whole. However, whether an arc is monotone may depend on the ordering of the values of the variables that it uses. Sometimes, the choice of such an ordering is rather arbitrary. In these cases, it is desirable to order the values of these variables such that all arcs (or as many arcs as possible) are monotone. In this paper we discuss the concept of local monotonicity and its computational complexity. We present an algorithm for determining whether there exists an ordering of the values of the variables such that all arcs in a network are monotone, and show that this can be done in time, exponential only in the treewidth of the network. On the other hand, optimizing the number of monotone arcs is NP-complete and hard to approximate as well. We sketch a branch-and-bound exact algorithm to find an optimal solution for this problem.