A polynomial approximation algorithm for the minimum fill-in problem

Natanzon, Assaf; Shamir, Ron; Sharan, Roded

doi:10.1145/276698.276710

Cited by 48 publications

(58 citation statements)

References 24 publications

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“…For graphs of degree at most d, they obtained a better approximation factor O((nd + k) √ d log 4 n)/k). Natanzon et al [17] provided another type of approximation algorithms for Minimum Fill-in. For an input graph with a minimum fill-in of size k, their algorithm produces a fill-in of size at most 8k 2 , i.e., within a factor of 8k of optimal.…”

Section: That Ismentioning

confidence: 99%

“…The best known kernelization algorithm is due to Natanzon et al [17], which for a given instance (G, k) outputs in time O(k 2 nm) an instance (G , k ) such that k ≤ k, |V (G )| ≤ 2k 2 + 4k, and (G, k) is a YES instance if and only if (G , k ) is. Note that not every kernelization algorithm for fill-in in general graphs produces a sparse kernel, even if the input is a sparse graph.…”

Section: That Ismentioning

confidence: 99%

“…Note that not every kernelization algorithm for fill-in in general graphs produces a sparse kernel, even if the input is a sparse graph. For example, the algorithm of Natanzon et al [17], while reducing the number of vertices in the input graph G, introduces new edges. Thus the resulting kernel G can be very dense.…”

Section: That Ismentioning

confidence: 99%

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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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Section: That Ismentioning

confidence: 99%

Section: That Ismentioning

confidence: 99%

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Minimum Fill-in of Sparse Graphs: Kernelization and Approximation

2013

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“…This fact encouraged researchers to focus on various alternatives that are computationally more efficient, at the cost of optimality or generality. Examples of the approaches that have been attempted include approximation [36], restricted input [7,6,34,29,10,28], parameterization [12,26,23,15,33] and minimal completions [19,21,22,25,38,41]. Here we consider the last alternative.…”

Section: Introductionmentioning

confidence: 99%

Characterizing and Computing Minimal Cograph Completions

Lokshtanov

Mancini

Papadopoulos

Frontiers in Algorithmics

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A cograph completion of an arbitrary graph G is a cograph supergraph of G on the same vertex set. Such a completion is called minimal if the set of edges added to G is inclusion minimal. In this paper we present two results on minimal cograph completions. The first is a a characterization that allows us to check in linear time whether a given cograph completion is minimal. The second result is a vertex incremental algorithm to compute a minimal cograph completion H of an arbitrary input graph G in O(|V (H)| + |E(H)|) time.

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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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A polynomial approximation algorithm for the minimum fill-in problem

Cited by 48 publications

References 24 publications

Minimum Fill-in of Sparse Graphs: Kernelization and Approximation

Minimum Fill-in of Sparse Graphs: Kernelization and Approximation

Characterizing and Computing Minimal Cograph Completions

A Parameterized Algorithm for Chordal Sandwich

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