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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


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

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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)


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
  • Approximation Algorithms
  • Semidefinite Programming
  • Max-Cut
  • Max-Bisection


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