17 research outputs found
Tree decompositions with small cost
The f-cost of a tree decomposition ({Xi | i e I}, T = (I;F))
for a function f : N -> R+ is defined as EieI f(|Xi|). This measure
associates with the running time or memory use of some algorithms
that use the tree decomposition. In this paper we investigate the
problem to find tree decompositions of minimum f-cost.
A function f : N -> R+ is fast, if for every i e N: f(i+1) => 2*f(i).
We show that for fast functions f, every graph G has a tree decomposition
of minimum f-cost that corresponds to a minimal triangulation
of G; if f is not fast, this does not hold. We give polynomial time
algorithms for the problem, assuming f is a fast function, for graphs
that has a polynomial number of minimal separators, for graphs of
treewidth at most two, and for cographs, and show that the problem
is NP-hard for bipartite graphs and for cobipartite graphs.
We also discuss results for a weighted variant of the problem derived
of an application from probabilistic networks
An extended depth-first search algorithm for optimal triangulation of Bayesian networks
The junction tree algorithm is currently the most popular algorithm for exact inference on Bayesian networks. To improve the time complexity of the junction tree algorithm, we need to find a triangulation with the optimal total table size. For this purpose, Ottosen and Vomlel have proposed a depth-first search (DFS) algorithm. They also introduced several techniques to improve the DFS algorithm, including dynamic clique maintenance and coalescing map pruning. Nevertheless, the efficiency and scalability of that algorithm leave much room for improvement. First, the dynamic clique maintenance allows to recompute some cliques. Second, in the worst case, the DFS algorithm explores the search space of all elimination orders, which has size n!, where n is the number of variables in the Bayesian network. To mitigate these problems, we propose an extended depth-first search (EDFS) algorithm. The new EDFS algorithm introduces the following two techniques as improvements to the DFS algorithm: (1) a new dynamic clique maintenance algorithm that computes only those cliques that contain a new edge, and (2) a new pruning rule, called pivot clique pruning. The new dynamic clique maintenance algorithm explores a smaller search space and runs faster than the Ottosen and Vomlel approach. This improvement can decrease the overhead cost of the DFS algorithm, and the pivot clique pruning reduces the size of the search space by a factor of O(n2). Our empirical results show that our proposed algorithm finds an optimal triangulation markedly faster than the state-of-the-art algorithm does