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**Published in:** LIPIcs, Volume 308, 32nd Annual European Symposium on Algorithms (ESA 2024)

We consider connectivity augmentation problems in the Steiner setting, where the goal is to augment the edge-connectivity between a specified subset of terminal nodes.
In the Steiner Augmentation of a Graph problem (k-SAG), we are given a k-edge-connected subgraph H of a graph G. The goal is to augment H by including links from G of minimum cost so that the edge-connectivity between nodes of H increases by 1. This is a generalization of the Weighted Connectivity Augmentation Problem, in which only links between pairs of nodes in H are available for the augmentation.
In the Steiner Connectivity Augmentation Problem (k-SCAP), we are given a Steiner k-edge-connected graph connecting terminals R, and we seek to add links of minimum cost to create a Steiner (k+1)-edge-connected graph for R. Note that k-SAG is a special case of k-SCAP.
The results of Ravi, Zhang and Zlatin for the Steiner Tree Augmentation problem yield a (1.5+ε)-approximation for 1-SCAP and for k-SAG when k is odd [Ravi et al., 2023]. In this work, we give a (1 + ln{2} +ε)-approximation for the Steiner Ring Augmentation Problem (SRAP). This yields a polynomial time algorithm with approximation ratio (1 + ln{2} + ε) for 2-SCAP. We obtain an improved approximation guarantee for SRAP when the ring consists of only terminals, yielding a (1.5+ε)-approximation for k-SAG for any k.

Daniel Hathcock and Michael Zlatin. Approximation Algorithms for Steiner Connectivity Augmentation. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 67:1-67:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{hathcock_et_al:LIPIcs.ESA.2024.67, author = {Hathcock, Daniel and Zlatin, Michael}, title = {{Approximation Algorithms for Steiner Connectivity Augmentation}}, booktitle = {32nd Annual European Symposium on Algorithms (ESA 2024)}, pages = {67:1--67:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-338-6}, ISSN = {1868-8969}, year = {2024}, volume = {308}, editor = {Chan, Timothy and Fischer, Johannes and Iacono, John and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2024.67}, URN = {urn:nbn:de:0030-drops-211387}, doi = {10.4230/LIPIcs.ESA.2024.67}, annote = {Keywords: Approximation Algorithms, Steiner Connectivity, Network Design} }

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APPROX

**Published in:** LIPIcs, Volume 317, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)

We consider minimum time multicasting problems in directed and undirected graphs: given a root node and a subset of t terminal nodes, multicasting seeks to find the minimum number of rounds within which all terminals can be informed with a message originating at the root. In each round, the telephone model we study allows the information to move via a matching from the informed nodes to the uninformed nodes.
Since minimum time multicasting in digraphs is poorly understood compared to the undirected variant, we study an intermediate problem in undirected graphs that specifies a target k < t, and requires the only k of the terminals be informed in the minimum number of rounds. For this problem, we improve implications of prior results and obtain an Õ(t^{1/3}) multiplicative approximation. For the directed version, we obtain an additive Õ(k^{1/2}) approximation algorithm (with a poly-logarithmic multiplicative factor). Our algorithms are based on reductions to the related problems of finding k-trees of minimum poise (sum of maximum degree and diameter) and applying a combination of greedy network decomposition techniques and set covering under partition matroid constraints.

Daniel Hathcock, Guy Kortsarz, and R. Ravi. The Telephone k-Multicast Problem. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 21:1-21:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{hathcock_et_al:LIPIcs.APPROX/RANDOM.2024.21, author = {Hathcock, Daniel and Kortsarz, Guy and Ravi, R.}, title = {{The Telephone k-Multicast Problem}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)}, pages = {21:1--21:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-348-5}, ISSN = {1868-8969}, year = {2024}, volume = {317}, editor = {Kumar, Amit and Ron-Zewi, Noga}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.21}, URN = {urn:nbn:de:0030-drops-210148}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2024.21}, annote = {Keywords: Network Design, Multicast, Steiner Poise} }

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