A sufficiently fast algorithm for finding close to optimal junction trees

An algorithm is developed for finding a close to optimal junction tree of a given graph G. The algorithm has a worst case complexity 0 (c k n a) where a and c are constants, n is the number of vertices , and k is the size of the largest clique in a junction tree of Gi n which this size is minimized. The algorithm guaran­ tees that the logarithm of the size of the state space of the heaviest clique in the junction tree produced is less than a constant factor off tl ; e optimal value. When k = O(logn), our algorithm yields a polynomial inference algorithm fo r Bayesian networks. 1 Introduction All exact inference algorithms for the computation of a posterior probability in general Bayesian net­ works have two conceptual phases. One phase handles operations on the graphical structure itself and the other performs probabilistic computations; The junc­ tion tree algorithm [LS88] requires us to first find a "good" junction tree and then perform probabilistic computations on the junction tree and the method of conditioning [Pe86] requires to find a "good" loop cut­ set and then perform a calculation using the loop cut­ set. In [BG94], we offered an algorithm that finds a loop cutset for which the logarithm of the state space is guaranteed to be a constant factor off the optimal value. In this paper, we provide a similar optimization for the junction tree algorithm. We shall first restrict our discussion to networks for which all vertices have the same state space size and to the optimality criterion which we call cliquewidth. The cliquewidth of an undirected graph G is the size of the largest clique in a junction tree of G in which the size of the largest clique is minimized. A more common term is treewidth which is the cliquewidth minus L To date all methods in the AI and Statistics commu­ nities for finding a junction tree had no guarantee of performance and could perfo rm rather poorly when presented with an appropriate example. One algo­ rithm, due to Rose (1974), is as fo llows: repeatedly, select a vertex v with minimum number of neighbors N(v), delete v fr om the graph, and make a clique out of N (v). The resulting sequence of cliques creates a junction tree. This greedy algorithm minimizes the size of each clique as it is …