MAP Complexity Results and Approximation Methods

TL;DR

This paper explores the computational complexity of MAP in Bayesian networks, proving NP-completeness even in restricted cases, and introduces a local search and belief propagation framework for practical approximation.

Contribution

It establishes the NP-completeness of MAP and proposes a novel approximation framework combining local search with belief propagation.

Findings

MAP is NP-complete even for polytrees.

The proposed method provides accurate MAP estimates in complex networks.

MAP remains hard to approximate effectively, motivating the new framework.

Abstract

MAP is the problem of finding a most probable instantiation of a set of nvariables in a Bayesian network, given some evidence. MAP appears to be a significantly harder problem than the related problems of computing the probability of evidence Pr, or MPE a special case of MAP. Because of the complexity of MAP, and the lack of viable algorithms to approximate it,MAP computations are generally avoided by practitioners. This paper investigates the complexity of MAP. We show that MAP is complete for NP. We also provide negative complexity results for elimination based algorithms. It turns out that MAP remains hard even when MPE, and Pr are easy. We show that MAP is NPcomplete when the networks are restricted to polytrees, and even then can not be effectively approximated. Because there is no approximation algorithm with guaranteed results, we investigate best effort approximations. We introduce a generic MAP approximation framework. As one instantiation of it, we implement local search coupled with belief propagation BP to approximate MAP. We show how to extract approximate evidence retraction information from belief propagation which allows us to perform efficient local search. This allows MAP approximation even on networks that are too complex to even exactly solve the easier problems of computing Pr or MPE. Experimental results indicate that using BP and local search provides accurate MAP estimates in many cases.