Treewidth: Computational Experiments

Abstract

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Exploiting the Probability of Observation for Efficient Bayesian Network Inference

“…In 1984 Tarjan and Yannaskakis presented Maximum Cardinality Search (MCS) [45,33,32] as a simplified implementation of Lex-BFS and its applications for recognition of chordal graphs, testing acyclicity of hypergraphs, and how to selectively reduce an acyclic hypergraph to more efficiently compute database queries. The MCS algorithm is similar to Lex-BFS but uses a single integer as a vertex label instead of a string.…”

Efficient Graph Reachability Query Answering Using Tree Decomposition

“…Although for fixed w, linear time algorithms exist to solve the decision problem "treewidth ≤ w" [3], there is a huge hidden constant factor, which prevents it to be useful in practice. There exist many heuristics and approximation algorithms for determining the treewidth, unfortunately few of them can deal with graphs containing more than 1000 nodes [11]. (2) The second problem lies in the fact that even if the treewidth can be determined, it still can not be guaranteed that good performance will be obtained since the time complexity of most of the algorithms is exponential to the treewidth.…”

Generation of Tree Decompositions by Iterated Local Search

“…The concept of tree decompositions has been first introduced by Robertson and Seymour ([11]): Definition 1. (see [11], [9]) Let G = (V, E) be a graph. A tree decomposition of G is a pair (T, χ), where T = (I, F ) is a tree with node set I and edge set F , and χ = {χ i : i ∈ I} is a family of subsets of V , one for each node of T , such that 1. i∈I χ i = V , 2. for every edge (v, w) ∈ E, there is an i ∈ I with v ∈ χ i and w ∈ χ i , and 3. for all i, j, k ∈ I, if j is on the path from i to k in T , then χ i ∩ χ k ⊆ χ j .…”

“…The process of elimination continues until the triangulation H is obtained. A more detailed description of the algorithm for constructing a graph's triangulation for a given elimination ordering is found in [9]. The treewidth of a triangulated graph is equal to the largest clique of triangulated graph minus 1 ([5]).…”

“…Finding the treewidth of a graph G is equivalent to finding a triangulation G of G with minimum clique size. The last two lemmas can be found in [9].…”