Approximation hardness of dominating set problems in bounded degree graphs

Chlebík, Miroslav; Chlebíková, Janka

doi:10.1016/j.ic.2008.07.003

Cited by 155 publications

(86 citation statements)

References 17 publications

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“…Any co-bipartite graph is also trivially quasi-line, and, in turn, claw-free. Hence our result contrasts again with the complexity of Min Dominating Set, which is approximable within a factor of − 1 for -claw-free graph 3 for any fixed , as shown in [19] using a short argument.…”

Section: Co-bipartite Graphscontrasting

confidence: 71%

“…It is well-known that Min Set Cover is an NP-hard problem [33], and that it is even log-APX-hard [52] (whereas logarithmic factors are tractable [39]); this even holds for the special case of Min Dominating Set [19,35,36]. The same properties hold for Min Test Cover [24] (and by Proposition 4, using Reduction 2 this result transfers to Min Discriminating Code) and Min Id Code (see [6,43,56], for different proofs).…”

Section: Related Workmentioning

confidence: 99%

“…[25] for an online database, and [20,35,36] for surveys and summaries). The log-APX-completeness of Min Dominating Set is known to hold even for bipartite graphs and split graphs [19], however it does not hold for planar graphs or unit disk graphs (for which Min Dominating Set admits PTAS algorithms [5,38,50]) or in (bipartite) graphs of bounded maximum degree (at least 3), where it is APX-complete [19].…”

Section: Related Workmentioning

confidence: 99%

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Decision and approximation complexity for identifying codes and locating-dominating sets in restricted graph classes

Foucaud

2015

Journal of Discrete Algorithms

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An identifying code is a subset of vertices of a graph with the property that each vertex is uniquely determined (identified) by its nonempty neighbourhood within the identifying code. When only vertices out of the code are asked to be identified, we get the related concept of a locating-dominating set. These notions are closely related to a number of similar and well-studied concepts such as the one of a test cover. In this paper, we study the decision problems Identifying Code and Locating-Dominating Set (which consist in deciding whether a given graph admits an identifying code or a locating-dominating set, respectively, with a given size) and their minimization variants Minimum Identifying Code and Minimum LocatingDominating Set. These problems are known to be NP-hard, even when the input graph belongs to a number of specific graph classes such as planar bipartite graphs. Moreover, it is known that they are approximable within a logarithmic factor, but hard to approximate within any sub-logarithmic factor. We extend the latter result to the case where the input graph is bipartite, split or co-bipartite: both problems remain hard in these cases. Among other results, we also show that for bipartite graphs of bounded maximum degree (at least 3), the two problems are hard to approximate within some constant factor, a question which was open. We summarize all known results in the area, and we compare them to the ones for the related problem Dominating Set. In particular, our work exhibits important graph classes for which Dominating Set is efficiently solvable, but Identifying Code and Locating-Dominating Set are hard (whereas in all previous works, their complexity was the same). We also introduce graph classes for which the converse holds, and for which the complexities of Identifying Code and Locating-Dominating Set differ.

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Section: Co-bipartite Graphscontrasting

confidence: 71%

Section: Related Workmentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

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Decision and approximation complexity for identifying codes and locating-dominating sets in restricted graph classes

Foucaud

2015

Journal of Discrete Algorithms

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“…The dominating set problem in bounded degree networks was considered by Chlebik and Chlebikova [4], who derive explicit lower bounds on the approximation ratios of centralized solutions. While we are not aware of any previous work on distributed approximation of dominating sets in bounded degree networks, several related problems were considered in this setting.…”

Section: Modelmentioning

confidence: 99%

Deterministic Dominating Set Construction in Networks with Bounded Degree

Friedman

Kogan

2011

Distributed Computing and Networking

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Abstract. This paper considers the problem of calculating dominating sets in networks with bounded degree. In these networks, the maximal degree of any node is bounded by Δ, which is usually significantly smaller than n, the total number of nodes in the system. Such networks arise in various settings of wireless and peer-to-peer communication. A trivial approach of choosing all nodes into the dominating set yields an algorithm with the approximation ratio of Δ + 1. We show that any deterministic algorithm with non-trivial approximation ratio requires Ω(log * n) rounds, meaning effectively that no o(Δ)-approximation deterministic algorithm with a running time independent of the size of the system may ever exist. On the positive side, we show two deterministic algorithms that achieve log Δ and 2 log Δ-approximation in O(Δ 3 + log * n) and O(Δ 2 log Δ + log * n) time, respectively. These algorithms rely on coloring rather than node IDs to break symmetry.

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“…It follows from [7] that Dominating Set on general graphs can approximated to within roughly ln(∆(G) + 1), where ∆(G) is the maximum degree in the graph G. On the other hand, it is NP-hard to approximate Dominating Set in bipartite graphs of degree at most B within a factor of (ln B − c ln ln B), for some absolute constant c [8]. Note that a graph of degree at most B excludes K B+2 as a topological minor, and, hence, the hardness also applies to graphs excluding K h as a topological minor.…”

Section: Introduction For Dominating Setmentioning

confidence: 99%

Parameterized Complexity of Directed Steiner Tree on Sparse Graphs

Jones

Lokshtanov

Ramanujan

et al. 2013

Lecture Notes in Computer Science

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Abstract. We study the parameterized complexity of the directed variant of the classical Steiner Tree problem on various classes of directed sparse graphs. While the parameterized complexity of Steiner Tree parameterized by the number of terminals is well understood, not much is known about the parameterization by the number of non-terminals in the solution tree. All that is known for this parameterization is that both the directed and the undirected versions are W[2]-hard on general graphs, and hence unlikely to be fixed parameter tractable (FPT). The undirected Steiner Tree problem becomes FPT when restricted to sparse classes of graphs such as planar graphs, but the techniques used to show this result break down on directed planar graphs. In this article we precisely chart the tractability border for Directed Steiner Tree (DST) on sparse graphs parameterized by the number of non-terminals in the solution tree. Specifically, we show that the problem is fixed parameter tractable on graphs excluding a topological minor, but becomes W[2]-hard on graphs of degeneracy 2. On the other hand we show that if the subgraph induced by the terminals is required to be acyclic then the problem becomes FPT on graphs of bounded degeneracy. We further show that our algorithm achieves the best possible running time dependence on the solution size and degeneracy of the input graph, under standard complexity theoretic assumptions. Using the ideas developed for DST, we also obtain improved algorithms for Dominating Set on sparse undirected graphs. These algorithms are asymptotically optimal.

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Approximation hardness of dominating set problems in bounded degree graphs

Cited by 155 publications

References 17 publications

Decision and approximation complexity for identifying codes and locating-dominating sets in restricted graph classes

Decision and approximation complexity for identifying codes and locating-dominating sets in restricted graph classes

Deterministic Dominating Set Construction in Networks with Bounded Degree

Parameterized Complexity of Directed Steiner Tree on Sparse Graphs

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