Encyclopedia of Algorithms

2008 Edition
| Editors: Ming-Yang Kao

Max Cut

1994; Goemans, Williamson1995; Goemans, Williamson
  • Alantha Newman
Reference work entry
DOI: https://doi.org/10.1007/978-0-387-30162-4_219

Keywords and Synonyms

Maximum bipartite subgraph    

Problem Definition

Given an undirected edge-weighted graph, \( { G=(V,E) } \)

This is a preview of subscription content, log in to check access.

Recommended Reading

  1. 1.
    Andersson, G., Engebretsen, L., Håstad, J.: A new way to use semidefinite programming with applications to linear equations mod p. J. Algorithms 39, 162–204 (2001)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Arora, S., Rao, S., Vazirani, U.: Expander flows, geometric embeddings and graph partitioning. In: Proceedings of the 36th Annual Symposium on the Theory of Computing (STOC), Chicago 2004, pp. 222–231Google Scholar
  3. 3.
    Barahona, F.: On cuts and matchings in planar graphs. Math. Program. 60, 53–68 (1993)MathSciNetGoogle Scholar
  4. 4.
    Blum, A., Konjevod, G., Ravi, R., Vempala, S.: Semi-definite relaxations for minimum bandwidth and other vertex‐ordering problems. Theor. Comput. Sci. 235, 25–42 (2000)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Charikar, M., Guruswami, V., Wirth, A.: Clustering with qualitative information. In: Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science (FOCS), Boston 2003, pp. 524–533Google Scholar
  6. 6.
    Chor, B., Sudan, M.: A geometric approach to betweeness. SIAM J. Discret. Math. 11, 511–523 (1998)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Delorme, C., Poljak, S.: Laplacian eigenvalues and the maximum cut problem. Math. Program. 62, 557–574 (1993)MathSciNetGoogle Scholar
  8. 8.
    Delorme, C., Poljak, S.: The performance of an eigenvalue bound in some classes of graphs. Discret. Math. 111, 145–156 (1993). Also appeared in: Proceedings of the Conference on Combinatorics, Marseille, 1990MathSciNetzbMATHGoogle Scholar
  9. 9.
    Feige, U., Schechtman, G.: On the optimality of the random hyperplane rounding technique for MAX-CUT. Random Struct. Algorithms 20(3), 403–440 (2002)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Frieze, A., Jerrum, M.R.: Improved approximation algorithms for MAX-k-CUT and MAX BISECTION. Algorithmica 18, 61–77 (1997)MathSciNetGoogle Scholar
  11. 11.
    Goemans, M.X., Williamson, D.P.: Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. J. ACM 42, 1115–1145 (1995)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Goemans, M.X., Williamson, D.P.: Approximation algorithms for MAX-3-CUT and other problems via complex semidefinite programming. STOC 2001 Special Issue of J. Comput. Syst. Sci. 68, 442–470 (2004)Google Scholar
  13. 13.
    Grötschel, M., Lovász, L., Schrijver, A.: The ellipsoid method and its consequences in combinatorial optimization. Combinatorica 1, 169–197 (1981)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Grötschel, M., Lovász, L., Schrijver, A.: Geometric Algorithms and Combinatorial Optimization. Springer, Berlin (1988)zbMATHGoogle Scholar
  15. 15.
    Halperin, E., Zwick, U.: A unified framework for obtaining improved approximation algorithms for maximum graph bisection problems. Random Struct. Algorithms 20(3), 382–402 (2002)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Håstad, J.: Some optimal inapproximability results. J. ACM 48, 798–869 (2001)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Karger, D.R., Motwani, R., Sudan, M.: Improved graph coloring via semidefinite programming. J. ACM 45(2), 246–265 (1998)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Karloff, H.J.: How good is the Goemans‐Williamson MAX CUT algorithm? SIAM J. Comput. 29(1), 336–350 (1999)Google Scholar
  19. 19.
    Karp, R.M.: Reducibility Among Combinatorial Problems. In: Complexity of Computer Computations, pp. 85–104. Plenum Press, New York (1972)Google Scholar
  20. 20.
    Khot, S.: On the power of unique 2‑prover 1-round games. In: Proceedings of the 34th Annual Symposium on the Theory of Computing (STOC), Montreal 2002, pp. 767–775Google Scholar
  21. 21.
    Khot, S., Kindler, G., Mossel, E., O'Donnell, R.: Optimal inapproximability results for MAX CUT and other 2‑variable CSPs? In: Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science (FOCS), Rome 2004, pp. 146–154Google Scholar
  22. 22.
    de Klerk, E., Pasechnik, D., Warners, J.: On approximate graph colouring and MAX-k-CUT algorithms based on the θ function. J. Combin. Optim. 8(3), 267–294 (2004)zbMATHGoogle Scholar
  23. 23.
    Mahajan, R., Hariharan, R.: Derandomizing semidefinite programming based approximation algorithms. In: Proceedings of the 36th Annual IEEE Symposium on Foundations of Computer Science (FOCS), Milwaukee 1995, pp. 162–169Google Scholar
  24. 24.
    Mohar, B., Poljak, S.: Eigenvalues and the max-cut problem. Czechoslov Math. J. 40(115), 343–352 (1990)MathSciNetGoogle Scholar
  25. 25.
    Newman, A.: A note on polyhedral relaxations for the maximum cut problem (2004). Unpublished manuscriptGoogle Scholar
  26. 26.
    Poljak, S.: Polyhedral and eigenvalue approximations of the max-cut problem. Sets, Graphs and Numbers. Colloqiua Mathematica Societatis Janos Bolyai 60, 569–581 (1992)Google Scholar
  27. 27.
    Poljak, S., Rendl, F.: Node and edge relaxations of the max-cut problem. Comput. 52, 123–137 (1994)Google Scholar
  28. 28.
    Poljak, S., Rendl, F.: Nonpolyhedral relaxations of graph‐bisection problems. SIAM J. Opt. 5, 467–487 (1995)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Poljak, S., Rendl, F.: Solving the max-cut using eigenvalues. Discret. Appl. Math. 62(1–3), 249–278 (1995)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Poljak, S., Tuza, Z.: Maximum cuts and large bipartite subgraphs. DIMACS Ser. Discret. Math. Theor. Comput. Sci. 20, 181–244 (1995)Google Scholar
  31. 31.
    Sahni, S., Gonzalez, T.: P‑complete approximation problems. J. ACM 23(3), 555–565 (1976)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Swamy, C.: Correlation clustering: maximizing agreements via semidefinite programming. In: Proceedings of 15th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), New Orleans 2004, pp. 526–527Google Scholar
  33. 33.
    Trevisan, L., Sorkin, G., Sudan, M., Williamson, D.: Gadgets, approximation, and linear programming. SIAM J. Comput. 29(6), 2074–2097 (2000)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  • Alantha Newman
    • 1
  1. 1.Department of Algorithms and ComplexityMax-Planck Institute for Computer ScienceSaarbrückenGermany