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Stable Approximation Algorithms for Dominating Set and Independent Set

Authors Mark de Berg, Arpan Sadhukhan , Frits Spieksma

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Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
  • Mathematics of computing → Graph algorithms
Keywords
  • Dynamic algorithms
  • approximation algorithms
  • stability
  • dominating set
  • independent set

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Abstract

We study Dominating Set and Independent Set for dynamic graphs in the vertex-arrival model. We say that a dynamic algorithm for one of these problems is k-stable when it makes at most k changes to its output independent set or dominating set upon the arrival of each vertex. We study trade-offs between the stability parameter k of the algorithm and the approximation ratio it achieves. We obtain the following results.  
- We show that there is a constant ε^* > 0 such that any dynamic (1+ε^*)-approximation algorithm for Dominating Set has stability parameter Ω(n), even for bipartite graphs of maximum degree 4. 
- We present algorithms with very small stability parameters for Dominating Set in the setting where the arrival degree of each vertex is upper bounded by d. In particular, we give a 1-stable (d+1)²-approximation, and a 3-stable (9d/2)-approximation algorithm. 
- We show that there is a constant ε^* > 0 such that any dynamic (1+ε^*)-approximation algorithm for Independent Set has stability parameter Ω(n), even for bipartite graphs of maximum degree 3. 
- Finally, we present a 2-stable O(d)-approximation algorithm for Independent Set, in the setting where the average degree of the graph is upper bounded by some constant d at all times.

Cite As Get BibTex

Mark de Berg, Arpan Sadhukhan, and Frits Spieksma. Stable Approximation Algorithms for Dominating Set and Independent Set. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 275, pp. 27:1-27:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2023.27

Author Details

Mark de Berg
  • Department of Mathematics and Computer Science, TU Eindhoven, The Netherlands
Arpan Sadhukhan
  • Department of Mathematics and Computer Science, TU Eindhoven, The Netherlands
Frits Spieksma
  • Department of Mathematics and Computer Science, TU Eindhoven, The Netherlands

Funding

MdB, AS, and FS are supported by the Dutch Research Council (NWO) through Gravitation-grant NETWORKS-024.002.003.

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