Approximation Algorithms for Intersection Graphs

Kammer, Frank; Tholey, Torsten

doi:10.1007/s00453-012-9671-1

Cited by 25 publications

(26 citation statements)

References 34 publications

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“…This improves their approximation for MAXIMUM WEIGHTED INDEPENDENT SET, MINIMUM VERTEX COL-ORING, and MINIMUM CLIQUE PARTITION in t-interval graphs. For MAXI-MUM WEIGHTED INDEPENDENT SET and MINIMUM VERTEX COLOR-ING this reaches the best known ratio of t [4] in a simpler way, and for the other problems it improves the best known approximation ratios [18]. Then Kammer et al proved that MAXIMUM WEIGHTED CLIQUE can be 2k-approximated in k-perfectly orientable graphs.…”

Section: Discussionmentioning

confidence: 77%

“…The difference between the 4t-approximation of Kammer et al [18] and our tapproximation lies in two places. In their paper they proved that t-interval graphs are 2t-perfectly orientable, but following the lines of Theorem 1 one can see that those graphs are t-perfectly orientable.…”

Section: Discussionmentioning

confidence: 94%

“…This actually gives a polynomial exact algorithm for the MAXIMUM CLIQUE in 2-track graphs [19], as τ = 2 in this case. For the other cases, Kammer et al [18] greatly improved the approximation ratios from roughly t 2 /2 to 4t, using the new notion of k-perfect orientability. Actually, earlier observations by Alon [1] and by Bar-Yehuda et al [4] (about approximating the chromatic number of t-interval graphs) imply that the trivial algorithm (finding the point of the representation belonging to the maximum number of intervals) is a 2t-approximation algorithm.…”

Section: Approximation Algorithmsmentioning

confidence: 97%

“…As far as the third question is concerned, Kammer, Tholey and Voepel [18] have already presented an improved polynomial-time approximation algorithm that achieves an approximation ratio of 4t for t-interval graphs. In this paper (Section 3), we present a linear time 2t-approximation algorithm, and a polynomial time t-approximation algorithm for MAXIMUM WEIGHTED CLIQUE in t-interval graphs (and thus in t-track graphs), t-circular interval graphs, and t-circular track graphs.…”

Section: Introductionmentioning

confidence: 98%

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The Maximum Clique Problem in Multiple Interval Graphs (Extended Abstract)

Francis

Ochem

2012

Graph-Theoretic Concepts in Computer Science

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Abstract. Multiple interval graphs are variants of interval graphs where instead of a single interval, each vertex is assigned a set of intervals on the real line. We study the complexity of the MAXIMUM CLIQUE problem in several classes of multiple interval graphs. The MAXIMUM CLIQUE problem, or the problem of finding the size of the maximum clique, is known to be NP-complete for t-interval graphs when t ≥ 3 and polynomialtime solvable when t = 1. The problem is also known to be NP-complete in t-track graphs when t ≥ 4 and polynomial-time solvable when t ≤ 2. We show that MAXIMUM CLIQUE is already NP-complete for unit 2-interval graphs and unit 3-track graphs. Further, we show that the problem is APX-complete for 2-interval graphs, 3-track graphs, unit 3-interval graphs and unit 4-track graphs. We also introduce two new classes of graphs called t-circular interval graphs and t-circular track graphs and study the complexity of the MAXIMUM CLIQUE problem in them. On the positive side, we present a polynomial time t-approximation algorithm for WEIGHTED MAXIMUM CLIQUE on t-interval graphs, improving earlier work with approximation ratio 4t.

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Section: Discussionmentioning

confidence: 77%

Section: Discussionmentioning

confidence: 94%

Section: Approximation Algorithmsmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 98%

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The Maximum Clique Problem in Multiple Interval Graphs (Extended Abstract)

Francis

Ochem

2012

Graph-Theoretic Concepts in Computer Science

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“…(An in-tournament is defined similarly.) Following the terminology of Kammer and Tholey [19], we say that an orientation of a graph is 1-perfect if the out-neighborhood of every vertex induces a tournament, and that a graph is 1-perfectly orientable (1-p.o. for short) if it has a 1-perfect orientation.…”

Section: Introductionmentioning

confidence: 99%

Partial Characterizations of 1‐Perfectly Orientable Graphs

Hartinger

Milanič

2016

Journal of Graph Theory

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We study the class of 1-perfectly orientable graphs, that is, graphs having an orientation in which every out-neighborhood induces a tournament. 1-perfectly orientable graphs form a common generalization of chordal graphs and circular arc graphs. Even though they can be recognized in polynomial time, little is known about their structure. In this paper, we develop several results on 1-perfectly orientable graphs. In particular, we: (i) give a characterization of 1-perfectly orientable graphs in terms of edge clique covers, (ii) identify several graph transformations preserving the class of 1-perfectly orientable graphs, (iii) exhibit an infinite family of minimal forbidden induced minors for the class of 1-perfectly orientable graphs, and (iv) characterize the class of 1-perfectly orientable graphs within the classes of cographs and of cobipartite graphs. The class of 1-perfectly orientable co-bipartite graphs coincides with the class of co-bipartite circular arc graphs.

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Avoidable Vertices and Edges in Graphs

Beisegel

Chudnovsky

Gurvich

et al. 2019

Lecture Notes in Computer Science

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A vertex in a graph is simplicial if its neighborhood forms a clique. We consider three generalizations of the concept of simplicial vertices: avoidable vertices (also known as OCF -vertices), simplicial paths, and their common generalization avoidable paths, introduced here. We present a general conjecture on the existence of avoidable paths. If true, the conjecture would imply a result due to Ohtsuki, Cheung, and Fujisawa from 1976 on the existence of avoidable vertices, and a result due to Chvátal, Sritharan, and Rusu from 2002 the existence of simplicial paths. In turn, both of these results generalize Dirac's classical result on the existence of simplicial vertices in chordal graphs.We prove that every graph with an edge has an avoidable edge, which settles the first open case of the conjecture. We point out a close relationship between avoidable vertices in a graph and its minimal triangulations, and identify new algorithmic uses of avoidable vertices, leading to new polynomially solvable cases of the maximum weight clique problem in classes of graphs simultaneously generalizing chordal graphs and circular-arc graphs. Finally, we observe that the proved cases of the conjecture have interesting consequences for highly symmetric graphs: in a vertex-transitive graph every induced two-edge path closes to an induced cycle, while in an edge-transitive graph every three-edge path closes to a cycle and every induced three-edge path closes to an induced cycle. * An extended abstract of this work is

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Approximation Algorithms for Intersection Graphs

Cited by 25 publications

References 34 publications

The Maximum Clique Problem in Multiple Interval Graphs (Extended Abstract)

The Maximum Clique Problem in Multiple Interval Graphs (Extended Abstract)

Partial Characterizations of 1‐Perfectly Orientable Graphs

Avoidable Vertices and Edges in Graphs

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