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Approximation bounds for quadratic maximization and max-cut problems with semidefinite programming relaxation

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Abstract

In this paper, we consider a class of quadratic maximization problems. For a subclass of the problems, we show that the SDP relaxation approach yields an approximation solution with the worst-case performance ratio at least α = 0.87856 ⋯. In fact, the estimated worst-case performance ratio is dependent on the data of the problem with α being a uniform lower bound. In light of this new bound, we show that the actual worst-case performance ratio of the SDP relaxation approach (with the triangle inequalities added) is at least α + δ d if every weight is strictly positive, where δ d > 0 is a constant depending on the problem dimension and data.

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References

  1. 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  MATH  MathSciNet  Google Scholar 

  2. Alon N, Sudakov B, Zwick, U. Constructing worst case instances for semidefinite programming based approximation algorithms. SIAM J Discrete Math, 15: 58–72 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  3. Zwick U. Outward rotations: a tool for rounding solutions of semidefinite programming relaxations, with applications to MAX CUT and other problems. In: Proceedings of the 31st Annual ACM Symposium on Theory of Computing, 1999, 679–687

  4. Alon N, Sudakov B. Bipartite subgraphs and the smallest eigenvalue. Combin, Probab and Comput, 9: 1–12 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  5. Feige U, Schechtman G. On the optimality of the random hyperplane rounding technique for MAX CUT. Random Structures and Algorithms, 20(3): 403–440 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  6. Karloff H. How good is the Goemans-Williamson MAX CUT algorithm? SIAM J Comput, 29: 336–350 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  7. Khot S, Kindler G, Mossel E, O’Donnell R. Optimal inapproximability results for MAX-CUT and other two-variable CSPs? In: Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science, 2004, 146–154

  8. Håstad J. Some optimal inapproximability results. In: Proceedings of the 29th Annual ACM Symposium on Theory of Computing, 1997, 1–10

  9. Bertsimas D, Ye Y. Semidefinite relaxation, multivariate normal distributions, and other statistics. In: Du, D-Z, Pardalos P M, eds. Handbook of Combinatorial Optimization 3. Kluwer Academic Publishers, 1998, 1–19

  10. Goemans M X. Semidefinite programming in combinatorial optimization. Math Programming, 79: 143–161 (1997)

    MathSciNet  Google Scholar 

  11. Han Q, Ye Y, Zhang J. An improved rounding method and semidefinite programming relaxation for graph partition. Math Programming, 92: 509–535 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  12. Helmberg C. Semidefinite programming for combinatorial optimization. ZIB-Report 00-34, 2000

  13. Lovász L. Semidefinite programs and combinatorial optimization. Technical Report. Redmond, WA 98052: Microsoft Research, 2000

  14. Rendl F. Semidefinite programming and combinatorial optimization. Technical Report. Austria: Universität Klagenfurt, Institut für Mathematik, 1998

    Google Scholar 

  15. Xu D, Ye Y, Zhang J. Approximating the 2-catalog segmentation problem using semidefinite programming relaxation. Optim Method and Softw, 18: 705–719 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  16. Nemirovski A, Roos C, Terlaky T. On the maximization of quadratic form over intersection of ellipsoids with common center. Math Programming, 86: 463–474 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  17. Nesterov Yu. Semidefinite relaxation and nonconvex quadratic optimization. Optim Methods and Softw, 9: 141–160 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  18. Nesterov Yu. Global quadratic optimization via conic relaxation. CORE Discussion Paper. Belgium: Louvain-la-Neuve, 1998

    Google Scholar 

  19. Tseng P. Further Results approximating nonconvex quadratic optimization by semidefinite programming relaxation. SIAM J Optim, 2003, 14: 268–283 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  20. Ye Y. Approximating quadratic programming with bound and quadratic constraints. Math Programming, 84: 219–226 (1999)

    MATH  Google Scholar 

  21. Ye Y. Approximating global quadratic optimization with convex quadratic constraints. J Global Optim, 15: 1–17 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  22. Zhang S. Quadratic maximization and semidefinite relaxation. Math Programming, 87: 453–465 (2000)

    Article  MATH  Google Scholar 

  23. Sturm J F, Zhang S. On cones of nonnegative quadratic functions. Math Oper Res, 28: 246–267 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  24. Ye Y, Zhang S. New results on quadratic minimization. SIAM J Optim, 14: 245–267 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  25. Kim S, Kojima M. Exact solutions of some nonconvex quadartic optimization problems via SDP and SOCP relaxations. Working Papepr. Todyo: Department of Mathematical and Information Sciences, The Tokyo Institute of Technology, 2001

    Google Scholar 

  26. Feige U, Langberg M. The RPR2 rounding technique for semidefinite programs. In: Proceedings of the 28th International Colloquium on Automata, Languages and Programming. Greece: Crete, 2001, 213–224

    Google Scholar 

  27. Feige U, Karpinski M, Langberg M. Improved approximation of max-cut on graphs of bounded degree. J Algorithms, 43: 201–219 (2002)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Applied Mathematics, Beijing University of Technology, Beijing, 100022, China

    Da-chuan Xu

  2. Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong, Shatin, Hong Kong, China

    Shu-zhong Zhang

Authors
  1. Da-chuan Xu
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  2. Shu-zhong Zhang
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Corresponding author

Correspondence to Da-chuan Xu.

Additional information

This work was supported by the National Natural Science Foundation of China (Grant No. 10401038) and Startup Grant for Doctoral Research of Beijing University of Technology and Hong Kong RGC Earmarked Grant CUHK4242/04E

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Xu, Dc., Zhang, Sz. Approximation bounds for quadratic maximization and max-cut problems with semidefinite programming relaxation. SCI CHINA SER A 50, 1583–1596 (2007). https://doi.org/10.1007/s11425-007-0080-x

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  • DOI: https://doi.org/10.1007/s11425-007-0080-x

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