We consider the Max--Section problem, where we are given an undirected graph G=(V,E)equipped with non-negative edge weights w: E → R_+ and the goal is to find a partition of V into three equisized parts while maximizing the total weight of edges crossing between different parts. Max-3-Section is closely related to other well-studied graph partitioning problems, e.g., Max-Cut, Max-3-Cut, and Max-Bisection. We present a polynomial time algorithm achieving an approximation of 0.795, that improves upon the previous best known approximation of 0.673. The requirement of multiple parts that have equal sizes renders Max-3-Section much harder to cope with compared to, e.g., Max-Bisection. We show a new algorithm that combines the existing approach of Lassere hierarchy along with a random cut strategy that suffices to give our result.
@InProceedings{katzelnick_et_al:LIPIcs.ESA.2023.69, author = {Katzelnick, Dor and Pillai, Aditya and Schwartz, Roy and Singh, Mohit}, title = {{An Improved Approximation Algorithm for the Max-3-Section Problem}}, booktitle = {31st Annual European Symposium on Algorithms (ESA 2023)}, pages = {69:1--69:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-295-2}, ISSN = {1868-8969}, year = {2023}, volume = {274}, editor = {G{\o}rtz, Inge Li and Farach-Colton, Martin and Puglisi, Simon J. 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.2023.69}, URN = {urn:nbn:de:0030-drops-187229}, doi = {10.4230/LIPIcs.ESA.2023.69}, annote = {Keywords: Approximation Algorithms, Semidefinite Programming, Max-Cut, Max-Bisection} }
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