Minimal triangulations of graphs: A survey

Pinar Heggernes
2006 Discrete Mathematics  
Any given graph can be embedded in a chordal graph by adding edges, and the resulting chordal graph is called a triangulation of the input graph. In this paper we study minimal triangulations, which are the result of adding an inclusion minimal set of edges to produce a triangulation. This topic was first studied from the standpoint of sparse matrices and vertex elimination in graphs. Today we know that minimal triangulations are closely related to minimal separators of the input graph. Since
more » ... e first papers presenting minimal triangulation algorithms appeared in 1976, several characterizations of minimal triangulations have been proved, and a variety of algorithms exist for computing minimal triangulations of both general and restricted graph classes. This survey presents and ties together these results in a unified modern notation, keeping an emphasis on the algorithms. Although we know today that minimal triangulations are closely related to minimal separators, sparse matrix computations was the first field to study different triangulations of a given graph [44, 75, 78] . Large sparse symmetric systems of equations arise in many areas of engineering, like the structural analysis of a car body, or the modeling of air flow around an airplane wing. The function to be computed can be often discretized as a mesh that covers the physical structure, where each point is connected to a few other points, and the related sparse matrix can simply be regarded as an adjacency matrix of this graph. Such systems are solved through standard methods of linear algebra, like Gaussian elimination followed by forward and backward substitution. However, during the elimination process nonzero entries, called fill, are inserted into cells of the matrix that originally held zeros, which increases the time needed to perform the elimination, the storage requirements, and the time needed to solve the system after the elimination. A graph corresponding to a sparse symmetric matrix A is a graph which has the nonzero structure of A as its adjacency matrix. A graph algorithm, known as Elimination Game [75] , was given in 1961, introducing the connection between sparse matrix computations and graphs. This algorithm simulates symmetric Gaussian elimination on graphs by repeatedly choosing a vertex v, adding edges to make the neighborhood of v into a clique, and then removing v from the graph. The edges that are added during Elimination Game correspond to the fill of Gaussian elimination, and they are called fill edges. The number of fill edges is heavily dependent on the order in which the vertices are processed. This ordering of the graph corresponds to the symmetric pivotal ordering of the rows and columns in Gaussian elimination. 1 The resulting triangulation is the graph of the filled sparse matrix resulting from Gaussian elimination. 2 An interesting connection is that the class of graphs produced by adding the fill edges of Elimination Game to the input graph is exactly the class of chordal graphs [40] . Thus symmetric Gaussian elimination and consequently Elimination Game correspond to computing triangulations of the given graph. As mentioned above, computing a triangulation with few edges is important for efficiently solving sparse systems of linear equations. Since the problem of computing minimum triangulations is NP-hard, the related polynomially solvable problem of computing minimal triangulations became interesting, and the first algorithms for it appeared in 1976 [72, 79] . The number of edges in a minimal triangulation can be far from minimum, as illustrated in Fig. 1 . Because of this there is a general belief that minimal triangulations are not interesting in practice for sparse matrix computations. However, minimal triangulations that contain low fill are indeed highly desirable for convenient storage of the sparse matrices that result from Gaussian elimination. For a given graph G, if the computed triangulation H is minimal, then subsequent perfect elimination orderings of H applied on G, which might be necessary for example for parallel computations, always result in the same triangulation H of G, and thus the allocated storage for the filled matrix is not disturbed, as we will explain in more detail later. Although the first introduced algorithms for minimal triangulations do not consider the number of fill edges, more recent minimal triangulation algorithms are able to produce minimal triangulations with low fill [10, 11, 13, 29, 76] . Such algorithms are useful also for the treewidth problem [17] . The first results and characterizations of minimal triangulations were given simultaneously by Ohtsuki [72], Ohtsuki et al. [73], and Rose et al. [79] in 1976. These results are strongly connected to vertex elimination. Algorithm LEX M 1 For many real applications, the sparse matrix at hand is positive-definite, so that numerical stability is ensured regardless of the pivotal order, and pivoting can be performed with the sole goal of reducing fill. 2 This matrix is not symmetric as it has only zeros below the diagonal. However, in this context it is considered a symmetric matrix, resulting from adding it to its transpose.
doi:10.1016/j.disc.2005.12.003 fatcat:3kyxcgffjja6zfkzucqoa6uwyu