On-line algorithms for the dominating set problem

King, Gow-Hsing; Tzeng, Wen-Guey

doi:10.1016/s0020-0190(96)00191-3

Cited by 11 publications

(9 citation statements)

References 4 publications

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“…Below is a lower bound of Ω( √ n) for planar bipartite graphs. We should mention that is strikingly similar to the lower bound on general graphs given in [10]. We provide a simple augmentation of their lower bound so that it not only consists of inputs that are revealed according to our model but inputs that are also planar bipartite graphs.…”

Section: Planar Bipartite Graphssupporting

confidence: 64%

“…In particular, we show that √ ∆ -DOMINATE is 3 √ ∆-competitive along with a lower bound of Ω( √ ∆) for any online algorithm, essentially closing the problem in our setting. As previously mentioned, the authors in [10] consider a setting similar to ours where their adversary is not required to reveal visible vertices and they assume that an algorithm has additional knowledge of input size n. In this setting they provide an algorithm that achieves competitive ratio of Θ( √ n) for arbitrary graphs. For the upper bound below we follow a proof nearly identical to theirs modulo some minor details and definitions.…”

Section: Graphs Of Bounded Degreementioning

confidence: 99%

“…Recall that the setting defined in [10] is nearly identical to ours except that an algorithm knows the input size n beforehand and the induced subgraph on the revealed vertices is not necessarily connected. Within this setting the authors establish a lower bound of Ω( √ n) for arbitrary graphs.…”

Section: Noncompetitive Graph Classesmentioning

confidence: 99%

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Online Domination: The Value of Getting to Know All your Neighbors

Harutyunyan¹,

Pankratov²,

Racicot³

2021

Preprint

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We study the dominating set problem in an online setting. An algorithm is required to guarantee competitiveness against an adversary that reveals the input graph one node at a time. When a node is revealed, the algorithm learns about the entire neighborhood of the node (including those nodes that have not yet been revealed). Furthermore, the adversary is required to keep the revealed portion of the graph connected at all times. We present an algorithm that achieves 2-competitiveness on trees and prove that this competitive ratio cannot be improved by any other algorithm. We also present algorithms that achieve 2.5-competitiveness on cactus graphs, (t − 1)-competitiveness on K1,t-free graphs, and Θ( √ ∆) for maximum degree ∆ graphs. We show that all of those competitive ratios are tight. Then, we study several more general classes of graphs, such as threshold, bipartite planar, and seriesparallel graphs, and show that they do not admit competitive algorithms (that is, when competitive ratio is independent of the input size). Previously, the dominating set problem was considered in a slightly different input model, where a vertex is revealed alongside its restricted neighborhood: those neighbors that are among already revealed vertices. Thus, conceptually, our results quantify the value of knowing the entire neighborhood at the time a vertex is revealed as compared to the restricted neighborhood. For instance, it was known in the restricted neighborhood model that 3-competitiveness is optimal for trees, whereas knowing the neighbors allows us to improve it to 2-competitiveness.

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Section: Planar Bipartite Graphssupporting

confidence: 64%

Section: Graphs Of Bounded Degreementioning

confidence: 99%

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Online Domination: The Value of Getting to Know All your Neighbors

Harutyunyan¹,

Pankratov²,

Racicot³

2021

Preprint

View full text Add to dashboard Cite

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“…So in such setting, if a node is made representative, its status cannot be changed later. Nevertheless, an on-line version of LBSC is very difficult; in fact, the competitive ratio of it is as worse as O(n − 1) [19]. The dynamic model that we propose makes local changes on the status of a node (whether it is a representative or not).…”

Section: Dynamic Simclus Algorithmmentioning

confidence: 99%

SimClus: an effective algorithm for clustering with a lower bound on similarity

2010

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Clustering algorithms generally accept a parameter k from the user, which determines the number of clusters sought. However, in many application domains, like document categorization, social network clustering, and frequent pattern summarization, the proper value of k is difficult to guess. An alternative clustering formulation that does not require k is to impose a lower bound on the similarity between an object and its corresponding cluster representative. Such a formulation chooses exactly one representative for every cluster and minimizes the representative count. It has many additional benefits. For instance, it supports overlapping clusters in a natural way. Moreover, for every cluster, it selects a representative object, which can be effectively used in summarization or semi-supervised classification task. In this work, we propose an algorithm, SimClus, for clustering with lower bound on similarity. It achieves a O(log n) approximation bound on the number of clusters, whereas for the best previous algorithm the bound can be as poor as O(n). Experiments on real and synthetic data sets show that our algorithm produces more than 40% fewer representative objects, yet offers the same or better clustering quality. We also propose a dynamic variant of the algorithm, which can be effectively used in an on-line setting.

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“…As the web graph is evolving, one wants to decide whether a new vertex is to be added to the already existing dominating set without recomputing the existing dominating set and with minimal computational effort. On-line strategies for the dominating set problem have been considered in the past [11,17] for general graphs. However the authors are not aware of any result on on-line algorithms for this problem in random graphs.…”

Section: Introductionmentioning

confidence: 99%

Dominating Sets in Web Graphs

Cooper

Klasing

Zito

2004

Algorithms and Models for the Web-Graph

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In this paper we study the size of generalised dominating sets in two graph processes which are widely used to model aspects of the world-wide web. On the one hand, we show that graphs generated this way have fairly large dominating sets (i.e. linear in the size of the graph). On the other hand, we present efficient strategies to construct small dominating sets.The algorithmic results represent an application of a particular analysis technique which can be used to characterise the asymptotic behaviour of a number of dynamic processes related to the web.

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On-line algorithms for the dominating set problem

Cited by 11 publications

References 4 publications

Online Domination: The Value of Getting to Know All your Neighbors

Online Domination: The Value of Getting to Know All your Neighbors

SimClus: an effective algorithm for clustering with a lower bound on similarity

Dominating Sets in Web Graphs

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