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**Published in:** LIPIcs, Volume 283, 34th International Symposium on Algorithms and Computation (ISAAC 2023)

Recently, many studies have been devoted to finding diverse solutions in classical combinatorial problems, such as Vertex Cover (Baste et al., IJCAI'20), Matching (Fomin et al., ISAAC'20) and Spanning Tree (Hanaka et al., AAAI'21). Finding diverse solutions is important in settings where the user is not able to specify all criteria of the desired solution. Motivated by an application in the field of system identification, we initiate the algorithmic study of k-Diverse Minimum s-t Cuts which, given a directed graph G = (V, E), two specified vertices s,t ∈ V, and an integer k > 0, asks for a collection of k minimum s-t cuts in G that has maximum diversity. We investigate the complexity of the problem for two diversity measures for a collection of cuts: (i) the sum of all pairwise Hamming distances, and (ii) the cardinality of the union of cuts in the collection. We prove that k-Diverse Minimum s-t Cuts can be solved in strongly polynomial time for both diversity measures via submodular function minimization. We obtain this result by establishing a connection between ordered collections of minimum s-t cuts and the theory of distributive lattices. When restricted to finding only collections of mutually disjoint solutions, we provide a more practical algorithm that finds a maximum set of pairwise disjoint minimum s-t cuts. For graphs with small minimum s-t cut, it runs in the time of a single max-flow computation. These results stand in contrast to the problem of finding k diverse global minimum cuts - which is known to be NP-hard even for the disjoint case (Hanaka et al., AAAI'23) - and partially answer a long-standing open question of Wagner (Networks 1990) about improving the complexity of finding disjoint collections of minimum s-t cuts.

Mark de Berg, Andrés López Martínez, and Frits Spieksma. Finding Diverse Minimum s-t Cuts. In 34th International Symposium on Algorithms and Computation (ISAAC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 283, pp. 24:1-24:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{deberg_et_al:LIPIcs.ISAAC.2023.24, author = {de Berg, Mark and L\'{o}pez Mart{\'\i}nez, Andr\'{e}s and Spieksma, Frits}, title = {{Finding Diverse Minimum s-t Cuts}}, booktitle = {34th International Symposium on Algorithms and Computation (ISAAC 2023)}, pages = {24:1--24:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-289-1}, ISSN = {1868-8969}, year = {2023}, volume = {283}, editor = {Iwata, Satoru and Kakimura, Naonori}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2023.24}, URN = {urn:nbn:de:0030-drops-193267}, doi = {10.4230/LIPIcs.ISAAC.2023.24}, annote = {Keywords: S-T MinCut, Diversity, Lattice Theory, Submodular Function Minimization} }

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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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