{"citation":{"short":"M. Henzinger, J. Li, S. Rao, D. Wang, in:, 35th Annual ACM-SIAM Symposium on Discrete Algorithms, Society for Industrial and Applied Mathematics, 2024, pp. 3089–3139.","apa":"Henzinger, M., Li, J., Rao, S., &#38; Wang, D. (2024). Deterministic near-linear time minimum cut in weighted graphs. In <i>35th Annual ACM-SIAM Symposium on Discrete Algorithms</i> (pp. 3089–3139). Alexandria, VA,  United States: Society for Industrial and Applied Mathematics. <a href=\"https://doi.org/10.1137/1.9781611977912.111\">https://doi.org/10.1137/1.9781611977912.111</a>","ista":"Henzinger M, Li J, Rao S, Wang D. 2024. Deterministic near-linear time minimum cut in weighted graphs. 35th Annual ACM-SIAM Symposium on Discrete Algorithms. SODA: Symposium on Discrete Algorithms, 3089–3139.","ieee":"M. Henzinger, J. Li, S. Rao, and D. Wang, “Deterministic near-linear time minimum cut in weighted graphs,” in <i>35th Annual ACM-SIAM Symposium on Discrete Algorithms</i>, Alexandria, VA,  United States, 2024, pp. 3089–3139.","mla":"Henzinger, Monika, et al. “Deterministic Near-Linear Time Minimum Cut in Weighted Graphs.” <i>35th Annual ACM-SIAM Symposium on Discrete Algorithms</i>, Society for Industrial and Applied Mathematics, 2024, pp. 3089–139, doi:<a href=\"https://doi.org/10.1137/1.9781611977912.111\">10.1137/1.9781611977912.111</a>.","ama":"Henzinger M, Li J, Rao S, Wang D. Deterministic near-linear time minimum cut in weighted graphs. In: <i>35th Annual ACM-SIAM Symposium on Discrete Algorithms</i>. Society for Industrial and Applied Mathematics; 2024:3089-3139. doi:<a href=\"https://doi.org/10.1137/1.9781611977912.111\">10.1137/1.9781611977912.111</a>","chicago":"Henzinger, Monika, Jason Li, Satish Rao, and Di Wang. “Deterministic Near-Linear Time Minimum Cut in Weighted Graphs.” In <i>35th Annual ACM-SIAM Symposium on Discrete Algorithms</i>, 3089–3139. Society for Industrial and Applied Mathematics, 2024. <a href=\"https://doi.org/10.1137/1.9781611977912.111\">https://doi.org/10.1137/1.9781611977912.111</a>."},"OA_place":"repository","article_processing_charge":"No","date_published":"2024-01-04T00:00:00Z","OA_type":"free access","status":"public","department":[{"_id":"MoHe"}],"project":[{"call_identifier":"H2020","_id":"bd9ca328-d553-11ed-ba76-dc4f890cfe62","grant_number":"101019564","name":"The design and evaluation of modern fully dynamic data structures"},{"grant_number":"Z00422","name":"Wittgenstein Award - Monika Henzinger","_id":"34def286-11ca-11ed-8bc3-da5948e1613c"},{"name":"Static and Dynamic Hierarchical Graph Decompositions","grant_number":"I05982","_id":"bda196b2-d553-11ed-ba76-8e8ee6c21103"},{"grant_number":"P33775 ","name":"Fast Algorithms for a Reactive Network Layer","_id":"bd9e3a2e-d553-11ed-ba76-8aa684ce17fe"}],"publication_identifier":{"eisbn":["9781611977912"]},"month":"01","ec_funded":1,"publication":"35th Annual ACM-SIAM Symposium on Discrete Algorithms","year":"2024","external_id":{"arxiv":["2401.05627"]},"scopus_import":"1","date_updated":"2024-11-04T11:08:10Z","abstract":[{"text":"In 1996, Karger [Kar96] gave a startling randomized algorithm that finds a minimum-cut in a (weighted) graph in time O(m log3 n) which he termed near-linear time meaning linear (in the size of the input) times a polylogarthmic factor. In this paper, we give the first deterministic algorithm which runs in near-linear time for weighted graphs.\r\nPreviously, the breakthrough results of Kawarabayashi and Thorup [KT19] gave a near-linear time algorithm for simple graphs (which was improved to have running time O(m log2 n log log n) in [HRW20].) The main technique here is a clustering procedure that perfectly preserves minimum cuts. Recently, Li [Li21] gave an m1+o(1) deterministic minimum-cut algorithm for weighted graphs; this form of running time has been termed “almost-linear”. Li uses almost-linear time deterministic expander decompositions which do not perfectly preserve minimum cuts, but he can use these clusterings to, in a sense, “derandomize” the methods of Karger.\r\nIn terms of techniques, we provide a structural theorem that says there exists a sparse clustering that preserves minimum cuts in a weighted graph with o(1) error. In addition, we construct it deterministically in near linear time. This was done exactly for simple graphs in [KT19, HRW20] and with polylogarithmic error for weighted graphs in [Li21]. Extending the techniques in [KT19, HRW20] to weighted graphs presents significant challenges, and moreover, the algorithm can only polylogarithmically approximately preserve minimum cuts. A remaining challenge is to reduce the polylogarithmic-approximate clusterings to 1 + o(1/ log n)-approximate so that they can be applied recursively as in [Li21] over O(log n) many levels. This is an additional challenge that requires building on properties of tree-packings in the presence of a wide range of edge weights to, for example, find sources for local flow computations which identify minimum cuts that cross clusters.","lang":"eng"}],"quality_controlled":"1","page":"3089-3139","date_created":"2024-11-04T10:54:21Z","doi":"10.1137/1.9781611977912.111","corr_author":"1","day":"04","arxiv":1,"author":[{"id":"540c9bbd-f2de-11ec-812d-d04a5be85630","first_name":"Monika H","full_name":"Henzinger, Monika H","last_name":"Henzinger","orcid":"0000-0002-5008-6530"},{"last_name":"Li","full_name":"Li, Jason","first_name":"Jason"},{"last_name":"Rao","full_name":"Rao, Satish","first_name":"Satish"},{"first_name":"Di","last_name":"Wang","full_name":"Wang, Di"}],"publication_status":"published","title":"Deterministic near-linear time minimum cut in weighted graphs","type":"conference","publisher":"Society for Industrial and Applied Mathematics","oa_version":"Preprint","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","_id":"18503","conference":{"location":"Alexandria, VA,  United States","start_date":"2024-01-07","end_date":"2024-01-10","name":"SODA: Symposium on Discrete Algorithms"},"acknowledgement":"This project has received funding from the European Research Council(ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant agreement No. 101019564 “The Design of Modern Fully Dynamic Data Structures (MoDyn-Struct)” and the Austrian Science Fund (FWF) project Z 422-N, project “Static and Dynamic Hierarchical Graph Decompositions”, I 5982-N, and project “Fast Algorithms for a Reactive Network Layer (ReactNet)”, P33775-N, with additional funding from the netidee SCIENCE Stiftung, 2020–2024.","language":[{"iso":"eng"}]}