New Results for the MAP Problem in Bayesian Networks

TL;DR

This paper investigates the computational complexity of the MAP problem in Bayesian networks, proving its hardness even in simple cases, and introduces an efficient exact algorithm and approximation scheme for practical network classes.

Contribution

It extends complexity results for MAP, demonstrates inapproximability in simple networks, and proposes a new exact algorithm and FPTAS for networks with bounded treewidth and states.

Findings

MAP remains hard in simple network topologies

New exact algorithm is empirically fast for certain networks

A Fully Polynomial Time Approximation Scheme is developed

Abstract

This paper presents new results for the (partial) maximum a posteriori (MAP) problem in Bayesian networks, which is the problem of querying the most probable state configuration of some of the network variables given evidence. First, it is demonstrated that the problem remains hard even in networks with very simple topology, such as binary polytrees and simple trees (including the Naive Bayes structure). Such proofs extend previous complexity results for the problem. Inapproximability results are also derived in the case of trees if the number of states per variable is not bounded. Although the problem is shown to be hard and inapproximable even in very simple scenarios, a new exact algorithm is described that is empirically fast in networks of bounded treewidth and bounded number of states per variable. The same algorithm is used as basis of a Fully Polynomial Time Approximation Scheme for MAP under such assumptions. Approximation schemes were generally thought to be impossible for this problem, but we show otherwise for classes of networks that are important in practice. The algorithms are extensively tested using some well-known networks as well as random generated cases to show their effectiveness.