Approximation Algorithms for Unit Disk Graphs

Leeuwen, Erik Jan van

doi:10.1007/11604686_31

Cited by 30 publications

(19 citation statements)

References 34 publications

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“…3). This settles one of the main questions that motivated this paper: recent research [40][41][42] had shown that natural problems having an fptas ω exist, but their position in the hierarchy of approximable problems was hitherto unclear.…”

Section: Introductionmentioning

confidence: 70%

Structure of Polynomial-Time Approximation

Leeuwen

2011

Theory Comput Syst

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Approximation schemes are commonly classified as being either a polynomial-time approximation scheme (ptas) or a fully polynomial-time approximation scheme (fptas). To properly differentiate between approximation schemes for concrete problems, several subclasses have been identified: (optimum-)asymptotic schemes (ptas ∞ , fptas ∞ ), efficient schemes (eptas), and size-asymptotic schemes. We explore the structure of these subclasses, their mutual relationships, and their connection to the classic approximation classes. We prove that several of the classes are in fact equivalent. Furthermore, we prove the equivalence of eptas to so-called convergent polynomial-time approximation schemes. The results are used to refine the hierarchy of polynomial-time approximation schemes considerably and demonstrate the central position of eptas among approximation schemes.We also present two ways to bridge the hardness gap between asymptotic approximation schemes and classic approximation schemes. First, using notions from fixedparameter complexity theory, we provide new characterizations of when problems have a ptas or fptas. Simultaneously, we prove that a large class of problems (including all MAX-SNP-complete problems) cannot have an optimum-asymptotic approximation scheme unless P = NP, thus strengthening results of Arora et al. (J. ACM 45(3):501-555, 1998). Secondly, we distinguish a new property exhibited by many optimization problems: pumpability. With this notion, we considerably generalize several problem-specific approaches to improve the effectiveness of approximation schemes with asymptotic behavior.

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Section: Introductionmentioning

confidence: 70%

Structure of Polynomial-Time Approximation

Leeuwen

2011

Theory Comput Syst

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“…Examples are: planar graphs (of course); graphs of intersections of objects in the plane with bounded objects density (disk graphs with bounded density (see, e.g., [16]) are a special case of these); and graphs with bounded degree and bounded treewidth. The latter can be shown using the tree decompositions implied by the results of Ding and Oporowski [7].…”

Section: Theorem 52 Crossing Parameter 1 Recognition In the Plane Imentioning

confidence: 99%

Algorithms for Graphs Embeddable with Few Crossings per Edge

Grigoriev

Bodlaender

2007

Algorithmica

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We consider graphs that can be embedded on a surface of bounded genus such that each edge has a bounded number of crossings. We prove that many optimization problems, including MAXIMUM INDEPENDENT SET, MINIMUM VERTEX COVER, MINIMUM DOMINATING SET and many others, admit polynomial time approximation schemes when restricted to such graphs. This extends previous results by Baker [1] and Eppstein [8] to a much broader class of graphs. We also prove that for the considered class of graphs, there are balanced separators of size O( √ n) where n is a number of vertices in the graph. On the negative side, we prove that it is intractable to recognize the graphs embeddable in the plane with at most one crossing per edge.

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“…A good survey of many generalizations of these results can be found in [29,31]. Minimum Vertex Cover and Maximum Independent Set in unit-disk graphs and general disk graphs were considered in [37,51,52]. Hochbaum and Maass, and later Chlebík and Chlebíková, considered intersection graphs of d-dimensional boxes in R d [18,35], while Erlebach et al [24] considered intersection graphs of general fat objects in the plane.…”

Section: Related Workmentioning

confidence: 99%

Minimum Vertex Cover in Rectangle Graphs

Bar-Yehuda

Hermelin

Rawitz

2010

Algorithms – ESA 2010

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This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and education use, including for instruction at the authors institution and sharing with colleagues.Other uses, including reproduction and distribution, or selling or licensing copies, or posting to personal, institutional or third party websites are prohibited. We consider the Minimum Vertex Cover problem in intersection graphs of axis-parallel rectangles on the plane. We present two algorithms: The first is an EPTAS for non-crossing rectangle families, rectangle families R where R 1 \ R 2 is connected for every pair of rectangles R 1 , R 2 ∈ R. This algorithm extends to intersection graphs of pseudo-disks. The second algorithm achieves a factor of (1.5 + ε) in general rectangle families, for any fixed ε > 0, and works also for the weighted variant of the problem. Both algorithms exploit the plane properties of axis-parallel rectangles in a novel way.

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Approximation Algorithms for Unit Disk Graphs

Cited by 30 publications

References 34 publications

Structure of Polynomial-Time Approximation

Structure of Polynomial-Time Approximation

Algorithms for Graphs Embeddable with Few Crossings per Edge

Minimum Vertex Cover in Rectangle Graphs

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