Embeddings of graphs into distributions of trees that preserve distances in expectation are a cornerstone of many optimization algorithms. Unfortunately, online or dynamic algorithms which use these embeddings seem inherently randomized and ill-suited against adaptive adversaries. In this paper we provide a new tree embedding which addresses these issues by deterministically embedding a graph into a single tree containing O(log n) copies of each vertex while preserving the connectivity structure of every subgraph and O(log² n)-approximating the cost of every subgraph. Using this embedding we obtain the first deterministic bicriteria approximation algorithm for the online covering Steiner problem as well as the first poly-log approximations for demand-robust Steiner forest, group Steiner tree and group Steiner forest.
@InProceedings{haepler_et_al:LIPIcs.ESA.2022.63, author = {Haepler, Bernhard and Hershkowitz, D. Ellis and Zuzic, Goran}, title = {{Adaptive-Adversary-Robust Algorithms via Small Copy Tree Embeddings}}, booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)}, pages = {63:1--63:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-247-1}, ISSN = {1868-8969}, year = {2022}, volume = {244}, editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva 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.2022.63}, URN = {urn:nbn:de:0030-drops-170016}, doi = {10.4230/LIPIcs.ESA.2022.63}, annote = {Keywords: Tree metrics, metric embeddings, approximation algorithms, group Steiner forest, group Steiner tree, demand-robust algorithms, online algorithms, deterministic algorithms} }
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