Complex Semidefinite Programming and Max-k-Cut

Author Alantha Newman

Thumbnail PDF


  • Filesize: 469 kB
  • 11 pages

Document Identifiers

Author Details

Alantha Newman

Cite AsGet BibTex

Alantha Newman. Complex Semidefinite Programming and Max-k-Cut. In 1st Symposium on Simplicity in Algorithms (SOSA 2018). Open Access Series in Informatics (OASIcs), Volume 61, pp. 13:1-13:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


In a second seminal paper on the application of semidefinite programming to graph partitioning problems, Goemans and Williamson showed in 2004 how to formulate and round a complex semidefinite program to give what is to date still the best-known approximation guarantee of .836008 for Max-3-Cut. (This approximation ratio was also achieved independently around the same time by De Klerk et al..) Goemans and Williamson left open the problem of how to apply their techniques to Max-k-Cut for general k. They point out that it does not seem straightforward or even possible to formulate a good quality complex semidefinite program for the general Max-k-Cut problem, which presents a barrier for the further application of their techniques. We present a simple rounding algorithm for the standard semidefinite programmming relaxation of Max-k-Cut and show that it is equivalent to the rounding of Goemans and Williamson in the case of Max-3-Cut. This allows us to transfer the elegant analysis of Goemans and Williamson for Max-3-Cut to Max-k-Cut. For k > 3, the resulting approximation ratios are about .01 worse than the best known guarantees. Finally, we present a generalization of our rounding algorithm and conjecture (based on computational observations) that it matches the best-known guarantees of De Klerk et al.
  • Graph Partitioning
  • Max-k-Cut
  • Semidefinite Programming


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Gunnar Andersson, Lars Engebretsen, and Johan Håstad. A new way of using semidefinite programming with applications to linear equations mod p. J. Algorithms, 39(2):162-204, 2001. URL:
  2. Etienne de Klerk, Dmitrii V. Pasechnik, and Joost P. Warners. On approximate graph colouring and max-k-cut algorithms based on the theta-function. J. Comb. Optim., 8(3):267-294, 2004. URL:
  3. Alan M. Frieze and Mark Jerrum. Improved approximation algorithms for MAX k-cut and MAX BISECTION. Algorithmica, 18(1):67-81, 1997. URL:
  4. Michel X. Goemans and David P. Williamson. Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. J. ACM, 42(6):1115-1145, 1995. URL:
  5. Michel X. Goemans and David P. Williamson. Approximation algorithms for m_ax-3-c_ut and other problems via complex semidefinite programming. J. Comput. Syst. Sci., 68(2):442-470, 2004. URL:
  6. Ai-fan Ling. Approximation algorithms for max 3-section using complex semidefinite programming relaxation. In Ding-Zhu Du, Xiaodong Hu, and Panos M. Pardalos, editors, Combinatorial Optimization and Applications, Third International Conference, COCOA 2009, Huangshan, China, June 10-12, 2009. Proceedings, volume 5573 of Lecture Notes in Computer Science, pages 219-230. Springer, 2009. URL:
  7. Konstantin Makarychev and Alantha Newman. Complex semidefinite programming revisited and the assembly of circular genomes. In Bernard Chazelle, editor, Innovations in Computer Science - ICS 2010, Tsinghua University, Beijing, China, January 7-9, 2011. Proceedings, pages 444-459. Tsinghua University Press, 2011. URL:
  8. Shuzhong Zhang and Yongwei Huang. Complex quadratic optimization and semidefinite programming. SIAM Journal on Optimization, 16(3):871-890, 2006. URL:
  9. Uri Zwick. Outward rotations: A tool for rounding solutions of semidefinite programming relaxations, with applications to MAX CUT and other problems. In Jeffrey Scott Vitter, Lawrence L. Larmore, and Frank Thomson Leighton, editors, Proceedings of the Thirty-First Annual ACM Symposium on Theory of Computing, May 1-4, 1999, Atlanta, Georgia, USA, pages 679-687. ACM, 1999. URL:
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail