Possibilistic Networks: MAP Query and Computational Analysis

Abstract

Possibilistic networks are powerful graphical uncertainty representations based on possibility theory. This paper analyzes the computational complexity of querying min-based and product-based possibilistic networks. It particularly focuses on a very common kind of queries: computing maximum a posteriori explanation (MAP). The main result of the paper is to show that the decision problem of answering MAP queries in both min-based and product-based possibilistic networks is NPcomplete. Such computational complexity results represent an advantage of possibilistic networks over probabilistic networks since MAP querying is NP PP-complete in probabilistic Bayesian networks. We provide the proof based on reduction from the 3SAT decision problem to MAP querying possibilistic networks decision problem. As well as reductions that are useful for implementation of MAP queries using SAT solvers.

A complexity analysis of MPE inference in possibilistic networks

“…) Simply put, the conditioning rule of possibility theory is not required to compute the maximum a posteriori assignment. In this section, we provide the full proof of Equation (7) of the above statement, stated in [2].…”

“…The complexity of MAP querying a min-based possibilistic network has already been discussed in [2] and it has been shown that MAP inference in this context is NP-complete.…”

“…In [2], it is stated that in the case of a MAP query, the problem can be reduced to finding the most plausible assignment of query variables Q compatible with the evidence e. More precisely, it can be rewritten as:…”

“…In [2], the authors analysed the computational complexity of MAP queries in min-based and product-based possibilistic networks. They showed that the decision problem behind MAP querying is NP-complete for both min-based and productbased possibilistic networks.…”

“…Regarding MAP querying a product-based possibilistic network, only the proof of NPhardness has been provided. The first part of this paper provides the proof of NP-completeness theorem, stated in [2], of MAP querying a product-based possibilistic network. In the second part of the paper, we address the complete analysis of the decision problems associated with the most plausible explanation (MPE) task in both min-based and product-based possibilistic networks, show using a reduction to the SAT decision problem, that it is NP-complete.…”