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An Improved Approximation Algorithm for the Max-3-Section Problem

Authors Dor Katzelnick , Aditya Pillai , Roy Schwartz, Mohit Singh

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Author Details

Dor Katzelnick
  • The Henry and Marilyn Taub Faculty of Computer Science, Technion, Haifa, Israel
Aditya Pillai
  • H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA, USA
Roy Schwartz
  • The Henry and Marilyn Taub Faculty of Computer Science, Technion, Haifa, Israel
Mohit Singh
  • H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA, USA

Funding

Mohit Singh and Aditya Pillai were supported in part by NSF CCF-2106444 and NSF CCF-1910423. Dor Katzelnick and Roy Schwartz receive funding from the European Union’s Horizon 2020 research and innovation program under grant agreement no. 852870-ERC-SUBMODULAR.

Acknowledgements

The authors would like to thank Uri Zwick for insightful discussions.

Cite As Get BibTex

Dor Katzelnick, Aditya Pillai, Roy Schwartz, and Mohit Singh. An Improved Approximation Algorithm for the Max-3-Section Problem. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 69:1-69:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.ESA.2023.69

Abstract

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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Approximation algorithms analysis
  • Theory of computation → Semidefinite programming
Keywords
  • Approximation Algorithms
  • Semidefinite Programming
  • Max-Cut
  • Max-Bisection

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