In the Min k-Cut problem, the input is a graph G and an integer k. The task is to find a partition of the vertex set of G into k parts, while minimizing the number of edges that go between different parts of the partition. The problem is NP-complete, and admits a simple 3ⁿ⋅n^𝒪(1) time dynamic programming algorithm, which can be improved to a 2ⁿ⋅n^𝒪(1) time algorithm using the fast subset convolution framework by Björklund et al. [STOC'07]. In this paper we give an algorithm for Min k-Cut with running time 𝒪((2-ε)ⁿ), for ε > 10^{-50}. This is the first algorithm for Min k-Cut with running time 𝒪(cⁿ) for c < 2.
@InProceedings{lokshtanov_et_al:LIPIcs.ICALP.2023.90, author = {Lokshtanov, Daniel and Saurabh, Saket and Surianarayanan, Vaishali}, title = {{Breaking the All Subsets Barrier for Min k-Cut}}, booktitle = {50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)}, pages = {90:1--90:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-278-5}, ISSN = {1868-8969}, year = {2023}, volume = {261}, editor = {Etessami, Kousha and Feige, Uriel and Puppis, Gabriele}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.90}, URN = {urn:nbn:de:0030-drops-181422}, doi = {10.4230/LIPIcs.ICALP.2023.90}, annote = {Keywords: Exact algorithms, min k-cut, exponential algorithms, graph algorithms, k-way cut} }
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