A practical algorithm for making filled graphs minimal

A wide-range algorithm for minimal triangulation from an arbitrary ordering

“…In order to combine the idea of small fill with minimal triangulations, Minimal Triangulation Sandwich Problem was introduced by Blair, Heggernes, and Telle [7]: Given (G, α), find a minimal triangulation H of G such that G ⊆ H ⊆ G + α . This approach enables the user to affect the produced fill by supplying a desired elimination ordering to the algorithm, while computing a triangulation which is minimal.…”

“…This approach enables the user to affect the produced fill by supplying a desired elimination ordering to the algorithm, while computing a triangulation which is minimal. In [7] the authors present an algorithm that removes fill edges from G + α in order to solve this problem. The complexity of their algorithm is O(f (m + f )), where f is the number of filled edges in the initial simplicial filled graph G + α , thus the algorithm works fast for elimination orderings resulting in low fill.…”

Treewidth: Characterizations, Applications, and Computations

“…Take a tree decomposition of the input graph G, obtained by some heuristic. Transform this into a triangulation H of G. If this is not a minimal triangulation, we can obtain a subgraph H of H that is a minimal triangulation of G, e.g., with the algorithm of [12]. The corresponding tree decomposition never has a larger treewidth compared to the first one, but sometimes has a smaller treewidth.…”

Characterizing and Computing Minimal Cograph Completions

“…Hence, if one is able to efficiently sample from the space of minimal completions, it is possible to pick the one in the sample with fewest fill edges and have good chances to produce a completion close to the minimum. This process, while only being a heuristic without any approximation guarantees, has proven to often be good enough for practical purposes [4,2]. In addition, the study of minimal completions gives a deep insight in the structure of the graph class we consider.…”

“…Hence, for polynomial time recognizable classes with this property, it becomes trivial to solve the characterization problem, and very easy to solve both the extraction and the computation problems as well. Examples of algorithms that exploit sandwich monotonicity for efficiently extracting and computing a minimal completion, are those for chordal [4], split [19], threshold and chain graph [22] completions. In contrast, among the classes that do not have the sandwich monotone property, the only one for which a solution to the characterization and extraction problems is known, is the class of interval graphs [24].…”