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APPROX

**Published in:** LIPIcs, Volume 275, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)

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.

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)

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@InProceedings{deberg_et_al:LIPIcs.APPROX/RANDOM.2023.27, author = {de Berg, Mark and Sadhukhan, Arpan and Spieksma, Frits}, title = {{Stable Approximation Algorithms for Dominating Set and Independent Set}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)}, pages = {27:1--27:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-296-9}, ISSN = {1868-8969}, year = {2023}, volume = {275}, editor = {Megow, Nicole and Smith, Adam}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2023.27}, URN = {urn:nbn:de:0030-drops-188527}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2023.27}, annote = {Keywords: Dynamic algorithms, approximation algorithms, stability, dominating set, independent set} }

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**Published in:** LIPIcs, Volume 227, 18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022)

Let P be a set of points in ℝ^d (or some other metric space), where each point p ∈ P has an associated transmission range, denoted ρ(p). The range assignment ρ induces a directed communication graph G_{ρ}(P) on P, which contains an edge (p,q) iff |pq| ⩽ ρ(p). In the broadcast range-assignment problem, the goal is to assign the ranges such that G_{ρ}(P) contains an arborescence rooted at a designated root node and the cost ∑_{p ∈ P} ρ(p)² of the assignment is minimized.
We study the dynamic version of this problem. In particular, we study trade-offs between the stability of the solution - the number of ranges that are modified when a point is inserted into or deleted from P - and its approximation ratio. To this end we introduce the concept of k-stable algorithms, which are algorithms that modify the range of at most k points when they update the solution. We also introduce the concept of a stable approximation scheme, or SAS for short. A SAS is an update algorithm alg that, for any given fixed parameter ε > 0, is k(ε)-stable and that maintains a solution with approximation ratio 1+ε, where the stability parameter k(ε) only depends on ε and not on the size of P. We study such trade-offs in three settings.
- For the problem in ℝ¹, we present a SAS with k(ε) = O(1/ε). Furthermore, we prove that this is tight in the worst case: any SAS for the problem must have k(ε) = Ω(1/ε). We also present algorithms with very small stability parameters: a 1-stable (6+2√5)-approximation algorithm - this algorithm can only handle insertions - a (trivial) 2-stable 2-approximation algorithm, and a 3-stable 1.97-approximation algorithm.
- For the problem in 𝕊¹ (that is, when the underlying space is a circle) we prove that no SAS exists. This is in spite of the fact that, for the static problem in 𝕊¹, we prove that an optimal solution can always be obtained by cutting the circle at an appropriate point and solving the resulting problem in ℝ¹.
- For the problem in ℝ², we also prove that no SAS exists, and we present a O(1)-stable O(1)-approximation algorithm. Most results generalize to when the range-assignment cost is ∑_{p ∈ P} ρ(p)^{α}, for some constant α > 1. All omitted theorems and proofs are available in the full version of the paper [Mark de Berg et al., 2021].

Mark de Berg, Arpan Sadhukhan, and Frits Spieksma. Stable Approximation Algorithms for the Dynamic Broadcast Range-Assignment Problem. In 18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 227, pp. 15:1-15:21, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{deberg_et_al:LIPIcs.SWAT.2022.15, author = {de Berg, Mark and Sadhukhan, Arpan and Spieksma, Frits}, title = {{Stable Approximation Algorithms for the Dynamic Broadcast Range-Assignment Problem}}, booktitle = {18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022)}, pages = {15:1--15:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-236-5}, ISSN = {1868-8969}, year = {2022}, volume = {227}, editor = {Czumaj, Artur and Xin, Qin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2022.15}, URN = {urn:nbn:de:0030-drops-161756}, doi = {10.4230/LIPIcs.SWAT.2022.15}, annote = {Keywords: Computational geometry, online algorithms, broadcast range assignment, stable approximation schemes} }

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