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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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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.

Cite AsGet 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

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.

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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