Linear-Time Generation of Random Chordal Graphs

Şeker, Oylum; Heggernes, Pinar; Ekim, Tınaz; Taşkın, Z. Caner

doi:10.1007/978-3-319-57586-5_37

Cited by 9 publications

(16 citation statements)

References 20 publications

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“…After proving the correctness,we give extensive computational tests to demonstrate the kind of chordal graphs that our algorithm generates using each of the different subtree generation methods. Compared to our preliminary version [31], we propose more algorithms, provide implementation details and comprehensive experiments which confirm our findings. We compare our methods with one another and with existing test results; we also implement one of the earlier proposed methods and include this in our tests.…”

Section: Introductionsupporting

confidence: 71%

“…We need to evaluate the order of n i=1 |V (T i )| to analyze the running time of Algorithm Chordal-Gen. Let us mention at this point that in the preliminary version of this paper [31],…”

Section: Output As G the Intersection Graph Of {Vmentioning

confidence: 99%

“…This is a clear shortcoming for the field, and it was even mentioned as an important open task at a Dagstuhl Seminar [19]. Until some years ago, most of the algorithms tailored for chordal A preliminary version of this paper has appeared in the Proceedings of the 10th International Conference on Algorithms and Complexity, CIAC 2017 [31].…”

Section: Introductionmentioning

confidence: 99%

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Generation of random chordal graphs using subtrees of a tree

et al. 2022

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Chordal graphs form one of the most studied graph classes. Several graph problems that are NP-hard in general become solvable in polynomial time on chordal graphs, whereas many others remain NP-hard. For a large group of problems among the latter, approximation algorithms, parameterized algorithms, and algorithms with moderately exponential or sub-exponential running time have been designed. Chordal graphs have also gained increasing interest during the recent years in the area of enumeration algorithms. Being able to test these algorithms on instances of chordal graphs is crucial for understanding the concepts of tractability of hard problems on graph classes. Unfortunately, only few published papers give algorithms for generating chordal graphs. Even in these papers, only very few methods aim for generating a large variety of chordal graphs. Surprisingly, none of these methods is directly based on the \intersection of subtrees of a tree" characterization of chordal graphs. In this paper, we give an algorithm for generating chordal graphs, based on the characterization that a graph is chordal if and only if it is the intersection graph of subtrees of a tree. Upon generating a random host tree, we give and test various methods that generate subtrees of the host tree. We compare our methods to one another and to existing ones for generating chordal graphs. Our experiments show that one of our methods is able to generate the largest variety of chordal graphs in terms of maximal clique sizes. Moreover, two of our subtree generation methods result in an overall complexity of our generation algorithm that is the best possible time complexity for a method generating the entire node set of subtrees in an "intersection of subtrees of a tree" representation. The instances corresponding to the results presented in this paper, and also a set of relatively small-sized instances are made available online.

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Section: Introductionsupporting

confidence: 71%

“…We need to evaluate the order of n i=1 |V (T i )| to analyze the running time of Algorithm Chordal-Gen. Let us mention at this point that in the preliminary version of this paper [31],…”

Section: Output As G the Intersection Graph Of {Vmentioning

confidence: 99%

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Generation of random chordal graphs using subtrees of a tree

et al. 2022

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“…We need to evaluate the order of n i=1 |V (T i )| to analyze the running time of Algorithm Chordal-Gen. Let us explain at this point that in the preliminary version of this paper [28],…”

Section: Algorithm Chordalgenmentioning

confidence: 99%

Generation of random chordal graphs using subtrees of a tree

Şeker¹,

Heggernes²,

Ekim³

et al. 2018

Preprint

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Chordal graphs form one of the most studied graph classes. Several graph problems that are NP-hard in general become solvable in polynomial time on chordal graphs, whereas many others remain NP-hard. For a large group of problems among the latter, approximation algorithms, parameterized algorithms, and algorithms with moderately exponential or sub-exponential running time have been designed. Chordal graphs have also gained increasing interest during the recent years in the area of enumeration algorithms. Being able to test these algorithms on instances of chordal graphs is crucial for understanding the concepts of tractability of hard problems on graph classes. Unfortunately, only few published papers give algorithms for generating chordal graphs. Even in these papers, only very few methods aim for generating a large variety of chordal graphs. Surprisingly, none of these methods is directly based on the "intersection of subtrees of a tree" characterization of chordal graphs. In this paper, we give an algorithm for generating chordal graphs, based on the characterization that a graph is chordal if and only if it is the intersection graph of subtrees of a tree. Upon generating a random host tree, we give and test various methods that generate subtrees of the host tree. We compare our methods to one another and to existing ones for generating chordal graphs. Our experiments show that one of our methods is able to generate the largest variety of chordal graphs in terms of maximal clique sizes. Moreover, two of our subtree generation methods result in an overall complexity of our generation algorithm that is the best possible time complexity for a method generating the entire node set of subtrees in a "intersection of subtrees of a tree" representation. The instances corresponding to the results presented in this paper, and also a set of relatively small-sized instances are made available online.

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“…We generated the random chordal graph instances via the method given inŞeker et al [23]. This method is shown to generate the most varied chordal graphs in terms of sizes of maximal cliques and is based on a well-known characterization of chordal graphs that states a graph is chordal if and only if it is the intersection graph of subtrees of a tree [12].…”

Section: Random Chordal Graph Generationmentioning

confidence: 99%

A decomposition approach to solve the selective graph coloring problem in some perfect graph families

2018

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Graph coloring is the problem of assigning a minimum number of colors to all vertices of a graph such that no two adjacent vertices receive the same color. The selective graph coloring problem is a generalization of the standard graph coloring problem; given a graph with a partition of its vertex set into clusters, the objective is to choose exactly one vertex per cluster so that, among all possible selections, the number of colors necessary to color the vertices in the selection is minimum. This study focuses on a decomposition based exact solution framework for selective coloring in some perfect graph families: in particular, permutation, generalized split, and chordal graphs where the selective coloring problem is known to be NP-hard. Our method combines integer programming techniques and combinatorial algorithms for the graph classes of interest. We test our method on graphs with different sizes and densities, present computational results and compare them with solving an integer programming formulation of the problem by CPLEX, and a state-of-the art algorithm from the literature. Our computational experiments indicate that our decomposition approach significantly improves solution performance in low-density graphs, and regardless of edge-density in the class of chordal graphs. the number of colors necessary to color the vertices in the selection is minimum. In a graph where each cluster consists of a single vertex, SEL-COL becomes equivalent to the classical graph coloring problem [5]. SEL-COL, or partition coloring as it is alternatively called in the literature, is motivated by the wavelength routing and assignment problem, and was introduced in Li and Simha [17]. There, a telecommunication network, which is a collection of terminal nodes linked by optical fibers capable of carrying a certain number of wavelengths, and a set of source-destination node pairs are given. The problem is concerned with finding the minimum number of distinct wavelengths to assign to each route, that is, paths connecting the given source-destination pairs, such that no two routes that have a common link are assigned the same wavelength. In this setting, the set of all possible routes and pairs of routes possessing a common link correspond to vertices and edges respectively, and groups of routes connecting a given source-destination pair constitute the clusters in the host graph. Then, selection of one vertex per cluster achieves the goal of finding a route between each pair of terminals, and coloring of the selection delivers a proper wavelength to each route.In the standard graph coloring problem, assignments (colors) on the entities (vertices) are done without any regard to alternative choices for them. However, as the example applications reveal, there are cases where entities have their own set of feasible options (clusters) and thus should be allocated one among those alternatives. In such cases, where the basic graph coloring framework fails to suffice, SEL-COL bridges the gap by offering the required flexibility.SEL-COL ha...

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Linear-Time Generation of Random Chordal Graphs

Cited by 9 publications

References 20 publications

Generation of random chordal graphs using subtrees of a tree

Generation of random chordal graphs using subtrees of a tree

Generation of random chordal graphs using subtrees of a tree

A decomposition approach to solve the selective graph coloring problem in some perfect graph families

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