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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  1. Gunnar Andersson. An approximation algorithm for max p-section. In STACS 99: 16th Annual Symposium on Theoretical Aspects of Computer Science Trier, Germany, March 4-6, 1999 Proceedings, pages 237-247. Springer, 2002. Google Scholar
  2. Per Austrin, Siavosh Benabbas, and Konstantinos Georgiou. Better balance by being biased: A 0.8776-approximation for max bisection. ACM Trans. Algorithms, 13(1):2:1-2:27, 2016. Google Scholar
  3. Etienne De Klerk, Dmitrii Pasechnik, Renata Sotirov, and Cristian Dobre. On semidefinite programming relaxations of maximum k-section. Mathematical programming, 136(2):253-278, 2012. Google Scholar
  4. Etienne de Klerk, Dmitrii V Pasechnik, and Joost P Warners. On approximate graph colouring and max-k-cut algorithms based on the θ-function. Journal of Combinatorial Optimization, 8:267-294, 2004. Google Scholar
  5. Uriel Feige and Michael Langberg. The rpr2 rounding technique for semidefinite programs. Journal of Algorithms, 60(1):1-23, 2006. Google Scholar
  6. Alan Frieze and Mark Jerrum. Improved approximation algorithms for max k-cut and max bisection. Algorithmica, 18(1):67-81, 1997. Google Scholar
  7. Daya Ram Gaur, Ramesh Krishnamurti, and Rajeev Kohli. The capacitated max k-cut problem. Mathematical Programming, 115:65-72, 2008. Google Scholar
  8. Michel X Goemans and David Williamson. Approximation algorithms for max-3-cut and other problems via complex semidefinite programming. In Proceedings of the thirty-third annual ACM symposium on Theory of computing, pages 443-452, 2001. Google Scholar
  9. Michel X Goemans and David P Williamson. Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. Journal of the ACM (JACM), 42(6):1115-1145, 1995. Google Scholar
  10. Eran Halperin and Uri Zwick. A unified framework for obtaining improved approximation algorithms for maximum graph bisection problems. Random Structures & Algorithms, 20(3):382-402, 2002. Google Scholar
  11. Richard M Karp. Reducibility among combinatorial problems. Complexity of Computer Computations, 1972. Google Scholar
  12. Subhash Khot, Guy Kindler, Elchanan Mossel, and Ryan O'Donnell. Optimal inapproximability results for MAX-CUT and other 2-variable csps? SIAM J. Comput., 37(1):319-357, 2007. Google Scholar
  13. Ai-fan Ling. Approximation algorithms for max 3-section using complex semidefinite programming relaxation. In Combinatorial Optimization and Applications: Third International Conference, COCOA 2009, Huangshan, China, June 10-12, 2009. Proceedings 3, pages 219-230. Springer, 2009. Google Scholar
  14. Alantha Newman. Complex semidefinite programming and max-k-cut. In SIAM Symposium on Simplicity in Algorithms, 2018. Google Scholar
  15. Prasad Raghavendra and Ning Tan. Approximating csps with global cardinality constraints using SDP hierarchies. In SODA, pages 373-387. SIAM, 2012. Google Scholar
  16. Yinyu Ye. A .699-approximation algorithm for max-bisection. Mathematical Programming, 90(1):101-111, March 2001. Google Scholar
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