The Computational Complexity of Monotonicity in Probabilistic Networks

Abstract

Many computational problems related to probabilistic networks are complete for complexity classes that have few ‘real world’ complete problems. For example, the decision variant of the inference problem (pr) is PP-complete, the map-problem is np $^{\textsc{pp}}$-complete and deciding whether a network is monotone in mode or distribution is co-np $^{\textsc{pp}}$-complete. We take a closer look at monotonicity; more specific, the computational complexity of determining whether the values of the variables in a probabilistic network can be ordered, such that the network is monotone. We prove that this problem – which is trivially co-$\textsc{np}^{\textsc{pp}}$-hard – is complete for the class co-$\textsc{np}^{\textsc{np}^{\textsc{pp}}}$ in networks which allow implicit representation.