Skip to main content
SpringerLink
Log in

Canonical dual approach to solving the maximum cut problem

  • Published:
Journal of Global Optimization Aims and scope Submit manuscript

Abstract

This paper presents a canonical dual approach for finding either an optimal or approximate solution to the maximum cut problem (MAX CUT). We show that, by introducing a linear perturbation term to the objective function, the maximum cut problem is perturbed to have a dual problem which is a concave maximization problem over a convex feasible domain under certain conditions. Consequently, some global optimality conditions are derived for finding an optimal or approximate solution. A gradient decent algorithm is proposed for this purpose and computational examples are provided to illustrate the proposed approach.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Barahona F.: Network design using cut inequalities. SIAM J. Optim. 6, 823–837 (1996)

    Article  Google Scholar 

  2. Barahona F., Grötschel M., Jünger M., Reinelt G.: An application of combinatorial optimization to statistical physics and circuit layout design. Oper. Res. 36, 493–513 (1988)

    Article  Google Scholar 

  3. Burer S., Monteir R.D.C.: A projected gradient algorithm for solving the maxcut SDP relaxation. Optim. Methods Softw. 15, 175–200 (2001)

    Article  Google Scholar 

  4. Caterina D.S.: The cut polytope and the boolean quadric polytope. Discret. Math. 79, 71–75 (1989)

    Google Scholar 

  5. De Simone C., Diehl M., Jiinger M., Mutzel P., Reinelt G., Rinaldi G.: Exact ground states of ising spin glasses: new experimental results with a branch-and-cut algorithm. J. Stat. Phys. 80, 487–496 (1995)

    Article  Google Scholar 

  6. Fang S.-C., Gao D.Y., Sheu R.-L., Wu S.-Y.: Canonical dual approach for solving 0-1 quadratic programming problems. J. Ind. Manag. Optim. 4, 125–142 (2008)

    Article  Google Scholar 

  7. Feng, J.-M., Lin, G.-X., Sheu, R.-L., Xia, Y.: Duality and solutions for quadratic programming over single non-homogeneous quadratic constraint. J. Glob. Optim. (in press) (2011)

  8. Gao D.Y.: Duality Principles in Nonconvex Systems: Theory, Methods and Applications. Kluwer, Dordrecht (2000)

    Google Scholar 

  9. Gao D.Y.: Perfect duality theory and complete set of solutions to a class of global optimization. Optimization 52, 467–493 (2003)

    Article  Google Scholar 

  10. Gao D.Y., Ruan N.: On the solutions to quadratic minimization problems with box and integer constraints. J. Glob. Optim. 47, 463–484 (2010)

    Article  Google Scholar 

  11. Gao D.Y., Sherali H.D.: Canonical duality theory: Connection between continuum mechanics and global optimization. In: Gao, D.Y., Sherali, H.D. (eds) Advances in Applied Mathematics Mathematics, pp. 257–326. Springer, Berlin (2009)

    Google Scholar 

  12. Gao, D.Y., Watson, L.T., Easterling, D.R., Thacker, W.I., Billups, S.C.: Solving the canonical dual of box- and integer-constrained nonconvex quadratic programs via a deterministic direct search algorithm. Optim. Methods Softw. (2011). doi:10.1080/10556788.2011.641125

  13. Goemans M.X., Williamson D.P.: Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. J. ACM 42, 1115–1145 (1995)

    Article  Google Scholar 

  14. Hammer P.L.: Some network flow problems solved with pseudo-boolean programming. Oper. Res. 13, 388–399 (1965)

    Article  Google Scholar 

  15. Karp R.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W. (eds) Complexity of Computer Computations, pp. 85–103. Plenum Press, New York (1972)

    Chapter  Google Scholar 

  16. Lu C., Wang Z., Xing W.: An improved lower bound and approximation algorithm for binary constrained quadratic programming problem. J. Glob. Optim. 48, 497–508 (2010)

    Article  Google Scholar 

  17. Lu C., Wang Z., Xing W., Fang S.-C.: Extended canonical duality and conic programming for solving 0-1 quadratic programming problems. J. Ind. Manag. Optim. 6, 779–793 (2010)

    Article  Google Scholar 

  18. Orlova G.I., Dorfman Y.G.: Finding the maximum cut in a graph. Eng. Cybern. 10, 502–506 (1972)

    Google Scholar 

  19. Reinei T.G.: TSPLIB—A traveling salesman problem library. ORSA J. Comput. 3, 376–384 (1991)

    Article  Google Scholar 

  20. Rockafellar R.T.: Conjugate Duality and Optimization. SIAM Publications, Philadelphia (1974)

    Book  Google Scholar 

  21. Rubinov A.M., Yang X.Q.: Lagrange-Type Functions in Constrained Non-Convex Optimization. Kluwer, Dordrecht (2003)

    Google Scholar 

  22. Sahni S., Gonzalez T.: P-complete approximation problems. J. ACM 23, 555–565 (1976)

    Article  Google Scholar 

  23. Wang Z., Fang S.-C., Gao D.Y., Xing W.: Global extremal conditions for multi-integer quadratic programming. J. Ind. Manag. Optim. 4, 213–225 (2008)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, Tsinghua University, Beijing, China

    Zhenbo Wang & Wenxun Xing

  2. Department of Industrial and Systems Engineering, North Carolina State University, Raleigh, NC, USA

    Shu-Cherng Fang

  3. School of Science, Information Technology and Engineering, University of Ballarat, Ballarat, VIC, Australia

    David Y. Gao

Authors
  1. Zhenbo Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shu-Cherng Fang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. David Y. Gao
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Wenxun Xing
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhenbo Wang.

Additional information

Wang’s research work has been supported by NSFC No. 10801087 and SRF for ROCS, SEM, Xing’s research work has been supported by NSFC No. 11171177 and the Key Project of Chinese Ministry of Education No. 108005, Fang’s research work has been supported by NSF No. DMI-0553310 and Gao’s research work has been supported by US Air Force (AFOSR) No. FA9550-10-1-0487.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Wang, Z., Fang, SC., Gao, D.Y. et al. Canonical dual approach to solving the maximum cut problem. J Glob Optim 54, 341–351 (2012). https://doi.org/10.1007/s10898-012-9881-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10898-012-9881-8

Keywords

Search

Navigation