Distributed Local Approximation of the Minimum k-Tuple Dominating Set in Planar Graphs

Czygrinow, Andrzej; Hańćkowiak, Michał; Szymańska, Edyta; Wawrzyniak, Wojciech; Witkowski, Marcin

doi:10.1007/978-3-319-14472-6_4

Cited by 14 publications

(28 citation statements)

References 10 publications

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“…Complexity : Constant. Notes : The paper improves the approximation ratio of Paper 7 [15] from 7 from to 6, making it closer to the 4 lower bound presented in the same paper.…”

Section: Improved Distributed Local Approximation Algorithm For Minim...supporting

confidence: 67%

Bibliography of distributed approximation beyond bounded degree

Feuilloley¹

2020

Preprint

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This document is an informal bibliography of the papers dealing with distributed approximation algorithms on graphs classes that are sparse, but not in the classic setting of bounded degree. We call these classes structurally sparse which means that they are sparse because of their geometric nature (planar, bounded genus and unit-disk graphs) and/or because they have bounded parameters (arboricity, expansion, growth, independence) or forbidden structures (forbidden minors). Introductory notesSome papers cited here deal with exact problems (not approximation problems), such as maximal independent set. These are included because of the strong connections between these exact problems and some technique used for approximation.We assume that the nodes do not have the knowledge of their locations. Some papers, especially in the literature about robots, assume such knowledge, and are not cited here.There has been series of papers improving one on the other, either generalizing on larger classes, or improving the approximation ratio. We chose to list the papers from the most recent to the oldest, to have the up-to-date results first. The bibliography generated at the end of this document is in alphabetical order to allow quick access to a specific reference. When citing a paper in the section of another paper, we first give the reference in term of section number and then as a pointer to the bibliography. Also, conference and journal versions sometimes differ, thus we cite all the versions. This document was first designed as a tool for personal research, but we decided to make it public as it seems it could be useful to others. It is not a very polished formal document, and we may have missed references, or not cited properly every paper. Please let us know if you find any mistake or omission. Graph classes consideredThe well-known graphs classes considered in this document are : bounded-expansion, planar, unit-disk, bounded-genus, bounded-arboricity, boundedindependence (a.k.a. bounded-growth) and minor-closed graphs. The relations between these classes are the following: Planar ⊆ bounded-genus ⊆ minor-closed ⊆ bounded expansion. Bounded-genus ⊆ bounded arboricity Unit-disk graphs ⊆ bounded-independence.

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“…Complexity : Constant. Notes : The paper improves the approximation ratio of Paper 7 [15] from 7 from to 6, making it closer to the 4 lower bound presented in the same paper.…”

Section: Improved Distributed Local Approximation Algorithm For Minim...supporting

confidence: 67%

Bibliography of distributed approximation beyond bounded degree

Feuilloley¹

2020

Preprint

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“…The second phase of our algorithm is inspired by results of Czygrinow et al [5] and the greedy domination algorithm for biclique-free graphs of [16]. Czygrinow et al [5] defined the notion of pseudo-covers, which provide a tool to carry out a fine grained analysis of vertices that can potentially belong to the sets A v used to dominate the red neighborhood N R pvq of a vertex v. This tool can in fact be applied to much more general graphs than planar graphs, namely, to all graphs that exclude some complete bipartite graph K t,t . A refined analysis for classes of bounded expansion was provided by Kublenz et al [10].…”

Section: Analyzing the Local Dominatorsmentioning

confidence: 99%

“…By defining the notion of pseudo-covers, Czygrinow et al [5] provided a tool to carry out a fine grained analysis of the vertices that can potentially dominate the remaining neighborhoods. Using ideas of [10] and [16] we provide an even finer analysis for planar graphs on which we base the second phase of our distributed algorithm and compute a second partial dominating set.…”

Section: Introductionmentioning

confidence: 99%

Local planar domination revisited

Ozan¹,

Siebertz²,

Vigny³

2021

Preprint

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We show how to compute a 20 -approximation of a minimum dominating set in a planar graph in a constant number of rounds in the LOCAL model of distributed computing. This improves on the previously best known approximation factor of 52, which was achieved by an elegant and simple algorithm of Lenzen et al. Our algorithm combines ideas from the algorithm of Lenzen et al. with recent work of Czygrinow et al. and Kublenz et al. to reduce to the case of bounded degree graphs, where we can simulate a distributed version of the classical greedy algorithm.

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“…Later Amiri et al [ASS16,AS16] provided a new analysis method to extend the result of Lenzen et al to bounded genus graphs. This has recently been improved to excluded minor graphs by Czygrinow et al [CHWW18].…”

Section: Related Workmentioning

confidence: 99%

“…On graphs of bounded expansion, there is only a logarithmic time constant factor approximation known for dominating set, however, it seems that one can extend the algorithm of [CHWW18] to bounded expansion graphs as they care only about local minors. If we go slightly beyond those graphs, to graphs of bounded arboricity (where every subgraph has a constant edge density), the situation is worse: only an O(log ∆)-approximation in O(log n) rounds is known.…”

Section: Related Workmentioning

confidence: 99%

Distributed Distance-$r$ Dominating Set on Sparse High-Girth Graphs

Amiri,

Wiederhake

2019

Preprint

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The dominating set problem and its generalization, the distance-r dominating set problem, are among the well-studied problems in the sequential settings. In distributed models of computation, unlike for domination, not much is known about distance-r domination. This is actually the case for other important closely-related covering problem, namely, the distance-r independent set problem.By result of Kuhn et al.[KMW16] we know the distributed domination problem is hard on high girth graphs; we study the problem on a slightly restricted subclass of these graphs: graphs of bounded expansion with high girth, i.e. their girth should be at least 4r + 3.We show that in such graphs, for every constant r, a simple greedy CONGEST algorithm provides a constant-factor approximation of the minimum distance-r dominating set problem, in a constant number of rounds. More precisely, our constants are dependent to r, not to the size of the graph. This is the first algorithm that shows there are non-trivial constant factor approximations in constant number of rounds for any distance r-covering problem in distributed settings.To show the dependency on r is inevitable, we provide an unconditional lower bound showing the same problem is hard already on rings. We also show that our analysis of the algorithm is relatively tight, that is any significant improvement to the approximation factor requires new algorithmic ideas.

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Distributed Local Approximation of the Minimum k-Tuple Dominating Set in Planar Graphs

Cited by 14 publications

References 10 publications

Bibliography of distributed approximation beyond bounded degree

Bibliography of distributed approximation beyond bounded degree

Local planar domination revisited

Distributed Distance-$r$ Dominating Set on Sparse High-Girth Graphs

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