Minimal triangulations of graphs: A survey

Heggernes, Pinar

doi:10.1016/j.disc.2005.12.003

Cited by 156 publications

(116 citation statements)

References 77 publications

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“…It arises in particular in sparse matrix computations [16] and in perfect phylogeny since it has the problem of triangulating colored graphs as a special case [1,2]. It can also be seen as a generalization of the problems of adding or deleting edges in a minimum or minimal way in an arbitrary input graph to obtain a chordal graph, which have attracted considerable attention [13,14,19,20,23,24,26,27]. The NP-completeness of the problem follows from the results of several papers [1,6,32].…”

Section: Introductionmentioning

confidence: 99%

A Parameterized Algorithm for Chordal Sandwich

Heggernes

Mancini

Nederlof

et al. 2010

Lecture Notes in Computer Science

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Given an arbitrary graph G = (V, E) and an additional set of admissible edges F , the Chordal Sandwich problem asks whether there exists a chordal graph (V, E ∪ F ) such that F ⊆ F . This problem arises from perfect phylogeny in evolution and from sparse matrix computations in numerical analysis, and it generalizes the widely studied problems of completions and deletions of arbitrary graphs into chordal graphs. As many related problems, Chordal Sandwich is NP-complete. In this paper we show that the problem becomes tractable when parameterized with a suitable natural measure on the set of admissible edges F . In particular, we give an algorithm with running time O(2 k n 5 ) to solve this problem, where k is the size of a minimum vertex cover of the graph (V, F ). Hence we show that the problem is fixed parameter tractable when parameterized by k. Note that the parameter does not assume any restriction on the input graph, and it concerns only the additional edge set F .

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Section: Introductionmentioning

confidence: 99%

A Parameterized Algorithm for Chordal Sandwich

Heggernes

Mancini

Nederlof

et al. 2010

Lecture Notes in Computer Science

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“…Besides sparse matrix computations, applications of Minimum Fill-in can be found in database management, artificial intelligence, and the theory of Bayesian statistics. The survey of Heggernes [15] gives an overview of techniques and applications of minimum and minimal triangulations.…”

Section: That Ismentioning

confidence: 99%

Minimum Fill-in of Sparse Graphs: Kernelization and Approximation

2013

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The Minimum Fill-in problem is to decide if a graph can be triangulated by adding at most k edges. The problem has important applications in numerical algebra, in particular in sparse matrix computations. We develop kernelization algorithms for the problem on several classes of sparse graphs. We obtain linear kernels on planar graphs, and kernels of size O(k 3/2 ) in graphs excluding some fixed graph as a minor and in graphs of bounded degeneracy. As a byproduct of our results, we obtain approximation algorithms with approximation ratios O(log k) on planar graphs and O( √ k log k) on H-minor-free graphs. These results significantly improve the previously known kernelization and approximation results for Minimum Fill-in on sparse graphs. ACM Subject Classification G.2.2 Graph Theory -Graph Algorithms Keywords and phrases IntroductionA graph is chordal (or triangulated) if every cycle of length at least four has a chord, i.e. an edge between nonadjacent vertices of the cycle. In the Minimum Fill-in problem (also known as Minimum Triangulation and Chordal Graph Completion) the task is to check if at most k edges can be added to a graph such that the resulting graph is chordal. That isMinimum Fill-in Input: A graph G = (V, E) and a non-negative integer k.This is a classical computational problem motivated by, and named after, a fundamental issue arising in sparse matrix computations. During Gaussian eliminations of large sparse matrices, new non-zero elements -called fill -can replace original zeros, thus increasing storage requirements, the time needed for the elimination, and the time needed to solve the system after the elimination. The problem of finding the right elimination ordering * The research of Fedor V.

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“…The results in Observation 4.1 and 4.2 follow from previous results on chordal graphs and minimal triangulations, but, because of their simplicity, we include a small proof anyway (for further references see [12]). …”

Section: Split+ke Graphsmentioning

confidence: 65%

“…Given a split+1v graph G with modulator {x}, we can check whether it is chordal in linear time [12] and, if so, output tw(G) = ω(G) − 1. Otherwise, if ω(G − x) + 1 = ω(G) we output tw(G) = ω(G − x) by Observation 4.10.…”

Section: Treewidthmentioning

confidence: 99%

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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 triangulations of graphs: A survey

Cited by 156 publications

References 77 publications

A Parameterized Algorithm for Chordal Sandwich

A Parameterized Algorithm for Chordal Sandwich

Minimum Fill-in of Sparse Graphs: Kernelization and Approximation

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

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