Minimal triangulation of a graph and optimal pivoting order in a sparse matrix

Ohtsuki, Tomi; Cheung, L.; Fujisawa, Toshio

doi:10.1016/0022-247x(76)90182-7

Cited by 63 publications

(57 citation statements)

References 7 publications

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“…An ordering α on G is called a perfect elimination ordering (peo) if G + α = G. Consequently, α is a peo of G + α . If G + α is a minimal triangulation of G, then α is called a minimal elimination ordering (meo) of G [22].…”

Section: Background and Motivationmentioning

confidence: 99%

“…In 1976 Ohtsuki, Cheung, and Fujisawa [22], and Rose, Tarjan, and Lueker [28] simultaneously and independently showed that a minimal triangulation can be found in polynomial time, presenting two different algorithms of O(nm) time for this purpose, where n is the number of vertices and m is the number of edges of the input graph G. No minimal triangulation algorithm has achieved a better time bound since these results. One of these algorithms, LEX M [28], has become one of the classical algorithms for minimal triangulation.…”

Section: Background and Motivationmentioning

confidence: 99%

“…Characterization 5.8 (Ohtsuki, Cheung, and Fujisawa [22]) An ordering α of the vertices of a graph G is a meo of G iff at each step i of the elimination game, for each pair {a, b} of non-adjacent vertices of…”

Section: Lb-triang Solves the Minimal Triangulation Sandwich Problemmentioning

confidence: 99%

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

Berry

Bordat

Heggernes

et al. 2006

Journal of Algorithms

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We present a new algorithm, called LB-Triang, which computes minimal triangulations. We give both a straightforward O(nm ) time implementation and a more involved O(nm) time implementation, thus matching the best known algorithms for this problem.Our algorithm is based on a process by Lekkerkerker and Boland for recognizing chordal graphs which checks in an arbitrary order whether the minimal separators contained in each vertex neighborhood are cliques. LB-Triang checks each vertex for this property and adds edges whenever necessary to make each vertex obey this property. As the vertices can be processed in any order, LB-Triang is able to compute any minimal triangulation of a given graph, which makes it significantly different from other existing triangulation techniques.We examine several interesting and useful properties of this algorithm, and give some experimental results. Background and motivationComputing a triangulation consists in embedding a given graph into a triangulated, or chordal, graph by adding a set of edges called a fill. If no proper subset of the fill can generate a chordal graph when added to the given graph, then this fill is said to be minimal, and the resulting chordal graph is called a minimal triangulation. The fill is said to be minimum if its cardinality is the smallest over all possible minimal fills, and the corresponding triangulation is called a minimum *

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Section: Background and Motivationmentioning

confidence: 99%

Section: Background and Motivationmentioning

confidence: 99%

See 1 more Smart Citation

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

Berry

Bordat

Heggernes

et al. 2006

Journal of Algorithms

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show abstract

“…Though some earlier work had been done on these problems ( [14,13]), the seminal paper is that of Rose, Tarjan and Lueker [15], which presented two very efficient algorithms to compute a peo or an meo. They introduced the concept of lexicographic order (which roughly speaking is a dictionary order), and used this for graph searches which at each step choose an unnumbered vertex of maximal label.…”

Section: Introductionmentioning

confidence: 99%

Maximal Label Search Algorithms to Compute Perfect and Minimal Elimination Orderings

Berry¹,

Krueger²,

Simonet³

2009

SIAM J. Discrete Math.

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Many graph search algorithms use a vertex labeling to compute an ordering of the vertices. We examine such algorithms which compute a peo (perfect elimination ordering) of a chordal graph, and corresponding algorithms which compute an meo (minimal elimination ordering) of a non-chordal graph, an ordering used to compute a minimal triangulation of the input graph.We express all known peo-computing search algorithms as instances of a generic algorithm called MLS (Maximal Label Search) and generalize Algorithm MLS into CompMLS, which can compute any peo.We then extend these algorithms to versions which compute an meo, and likewise generalize all known meo-computing search algorithms. We show that not all minimal triangulations can be computed by such a graph search, and, more surprisingly, that all these search algorithms compute the same set of minimal triangulations, even though the computed meos are different.Finally, we present a complexity analysis of these algorithms.

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“…Running the Elimination Game on a graph G = (V, E) and an ordering β of its vertices, means to remove the vertices from G in the order given by β so that, after removing a vertex, we make its neighborhood in the current graph into a clique. It is well known that this produces a triangulation of G. In particular for each minimal triangulation H of a graph G, there exists an ordering β that can produce it [24]. An ordering is called perfect elimination ordering if every vertex is simplicial when it is deleted during the Elimination Game.…”

Section: Split+ke Graphsmentioning

confidence: 99%

Minimum Fill-In and Treewidth of Split+ ke and Split+ kv Graphs

Mancini

Algorithms and Computation

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In this paper we investigate how graph problems that are NP-hard in general, but polynomially solvable on split graphs, behave on input graphs that are close to being split. For this purpose we define split+ke and split+kv graphs to be the graphs that can be made split by removing at most k edges and at most k vertices, respectively. We show that problems like treewidth and minimum fill-in are fixed parameter tractable with parameter k on split+ke graphs. Along with positive results of fixed parameter tractability of several problems on split+ke and split+kv graphs, we also show a surprising hardness result. We prove that computing the minimum fill-in of split+kv graphs is NP-hard even for k = 1. This implies that also minimum fill-in of chordal+kv graphs is NP-hard for every k. In contrast, we show that the treewidth of split+1v graphs can be computed in polynomial time. This gives probably the first graph class for which the treewidth and the minimum fill-in problems have different computational complexity.

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Minimal triangulation of a graph and optimal pivoting order in a sparse matrix

Cited by 63 publications

References 7 publications

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

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

Maximal Label Search Algorithms to Compute Perfect and Minimal Elimination Orderings

Minimum Fill-In and Treewidth of Split+ ke and Split+ kv Graphs

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