Approximation Algorithms for B 1-EPG Graphs

Epstein, Dror; Golumbic, Martin Charles; Morgenstern, Gila

doi:10.1007/978-3-642-40104-6_29

Cited by 26 publications

(31 citation statements)

References 10 publications

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“…On the other hand, every k-bend graph is also a (k + 1)-interval graph [13]). Bar-Yehuda et al [3] show a 2t-approximation algorithm for the Maximum Weighted Independent Set for t-interval graphs, which gives a 4-approximation for 1-bend graphs (this bound was later obtained independently by Epstein et al [10] for the unweighted problem). It would be interesting to improve the approximation ratio or prove a lower bound (under some well-established complexity assumption).…”

Section: Problemmentioning

confidence: 95%

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On edge intersection graphs of paths with 2 bends

Pergel

Rzążewski²

2017

Discrete Applied Mathematics

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An EPG-representation of a graph G is a collection of paths in a plane square grid, each corresponding to a single vertex of G, so that two vertices are adjacent if and only if their corresponding paths share infinitely many points. In this paper we focus on graphs admitting EPG-representations by paths with at most 2 bends. We show hardness of the recognition problem for this class of graphs, along with some subclasses.We also initiate the study of graphs representable by unaligned polylines, and by polylines, whose every segment is parallel to one of prescribed slopes. We show hardness of recognition and explore the trade-off between the number of bends and the number of slopes.

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Section: Problemmentioning

confidence: 95%

“…There are two main branches in this kind of research. The first one is understanding the structure of graphs with at most k bends -so far, the case of 1-bend graphs received most attention [12,8,10,2]. The other is finding the smallest k, such that every graph of a given class G is a k-bend graph.…”

Section: Introductionmentioning

confidence: 99%

On edge intersection graphs of paths with 2 bends

Pergel

Rzążewski²

2017

Discrete Applied Mathematics

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“…In particular, it implies that the number of maximal cliques in B 1 -EPG graphs is polynomial. Epstein et al [7] remarked that the representation of the graph is not needed since the neighborhood of every vertex is a weakly chordal graph. When the number of bends is at least 2, such a proof scheme cannot hold since there might be an exponential number of maximal cliques.…”

Section: Maximum Clique Problem On Epg Graphsmentioning

confidence: 99%

Computing Maximum Cliques in $$B_2$$ -EPG Graphs

Bousquet

Heinrich

2017

Graph-Theoretic Concepts in Computer Science

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EPG graphs, introduced by Golumbic et al. in 2009, are edge-intersection graphs of paths on an orthogonal grid. The class B k -EPG is the subclass of EPG graphs where the path on the grid associated to each vertex has at most k bends. Epstein et al. showed in 2013 that computing a maximum clique in B1-EPG graphs is polynomial. As remarked in , when the number of bends is at least 4, the class contains 2-interval graphs for which computing a maximum clique is an NP-hard problem. The complexity status of the Maximum Clique problem remains open for B2 and B3-EPG graphs. In this paper, we show that we can compute a maximum clique in polynomial time in B2-EPG graphs given a representation of the graph.Moreover, we show that a simple counting argument provides a 2(k + 1)-approximation for the coloring problem on B k -EPG graphs without knowing the representation of the graph. It generalizes a result of [Epstein et al, 2013] on B1-EPG graphs (where the representation was needed).

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“…[3,4,13,9]) or gave approximation algorithms for graphs with an EPG-representation with few bends (see e.g. [8,17]).…”

Section: Introductionmentioning

confidence: 99%

EPG-representations with Small Grid-Size

Biedl

Derka

Dujmović

et al. 2018

Lecture Notes in Computer Science

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In an EPG-representation of a graph G, each vertex is represented by a path in the rectangular grid, and (v, w) is an edge in G if and only if the paths representing v an w share a grid-edge. Requiring paths representing edges to be x-monotone or, even stronger, both x-and y-monotone gives rise to three natural variants of EPG-representations, one where edges have no monotonicity requirements and two with the aforementioned monotonicity requirements. The focus of this paper is understanding how small a grid can be achieved for such EPG-representations with respect to various graph parameters. We show that there are m-edge graphs that require a grid of area Ω(m) in any variant of EPG-representations. Similarly there are pathwidth-k graphs that require height Ω(k) and area Ω(kn) in any variant of EPGrepresentations. We prove a matching upper bound of O(kn) area for all pathwidth-k graphs in the strongest model, the one where edges are required to be both x-and y-monotone. Thus in this strongest model, the result implies, for example, O(n), O(n log n) and O(n 3/2 ) area bounds for bounded pathwidth graphs, bounded treewidth graphs and all classes of graphs that exclude a fixed minor, respectively. For the model with no restrictions on the monotonicity of the edges, stronger results can be achieved for some graph classes, for example an O(n) area bound for bounded treewidth graphs and O(n log 2 n) bound for graphs of bounded genus.

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Approximation Algorithms for B 1-EPG Graphs

Cited by 26 publications

References 10 publications

On edge intersection graphs of paths with 2 bends

On edge intersection graphs of paths with 2 bends

Computing Maximum Cliques in $$B_2$$ -EPG Graphs

EPG-representations with Small Grid-Size

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