Polynomially Solvable Cases of Binary Quadratic Programs

  • Duan LiEmail author
  • Xiaoling SunEmail author
  • Shenshen GuEmail author
  • Jianjun GaoEmail author
  • Chunli LiuEmail author
Part of the Springer Optimization and Its Applications book series (SOIA, volume 39)


We summarize in this chapter polynomially solvable subclasses of binary quadratic programming problems studied in the literature and report some new polynomially solvable subclasses revealed in our recent research. It is well known that the binary quadratic programming program is NP-hard in general. Identifying polynomially solvable subclasses of binary quadratic programming problems not only offers theoretical insight into the complicated nature of the problem but also provides platforms to design relaxation schemes for exact solution methods. We discuss and analyze in this chapter six polynomially solvable subclasses of binary quadratic programs, including problems with special structures in the matrix Q of the quadratic objective function, problems defined by a special graph or a logic circuit, and problems characterized by zero duality gap of the SDP relaxation. Examples and geometric illustrations are presented to provide algorithmic and intuitive insights into the problems.


binary quadratic programming polynomial solvability series-parallel graph logic circuit lagrangian dual SDP relaxation 



This research was partially supported by the Research Grants Council of Hong Kong under grant 414207 and by the National Natural Science Foundation of China under grants 70671064 and 70832002.


  1. 1.
    Allemand, K., Fukuda, K., Liebling, T.M., Steiner, E.: A polynomial case of unconstrained zero-one quadratic optimization. Math. Program. 91, 49–52 (2001)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Avis, D., Kukuda, K.: Reverse search for numeration. Discrete Appl. Math. 65, 21–46 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Barahona, F.: A solvable case of quadratic 0–1 programming. Discrete Appl. Math. 13, 23–26 (1986)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Barahona, F., Jünger, M., Reinelt, G.: Experiments in quadratic 0–1 programming. Math. Program. 44, 127–137 (1989)Google Scholar
  5. 5.
    Beck, A., Teboulle, M.: Global optimality conditions for quadratic optimization problems with binary constraints. SIAM J. Optim. 11, 179–188 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Ben-Tal, A.: Conic and Robust Optimization, Lecture Notes, Universita di Roma La Sapienzia, Rome, Italy (2002)Google Scholar
  7. 7.
    Billionnet, A., Elloumi, S.: Using a mixed integer quadratic programming solver for the unconstrained quadratic 0–1 problem. Math. Program. 109, 55–68 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Chaillou, P., Hansen, P., Mahieu, Y.: Best network flow bounds for the quadratic knapsack problem. Lect. Notes Math. 1403, 226–235 (1986)Google Scholar
  9. 9.
    Chakradhar, S.T., Bushnell, M.L.: A solvable class of quadratic 0–1 programming. Discrete Appl. Math. 36, 233–251 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Chardaire, P., Sutter, A.: A decomposition method for quadratic zero-one programming. Manage. Sci. 41, 704–712 (1995)zbMATHCrossRefGoogle Scholar
  11. 11.
    Crama, Y., Hansen, P., Jaumard, B.: The basic algorithm for pseudo-Boolean programming revisited. Discrete Appl. Math. 29, 171–185 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Delorme, C., Poljak, S.: Laplacian eigenvalues and the maximum cut problem. Math. Program. 62, 557–574 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Ferrez, J.A., Fukuda, K., Liebling, T.M.: Solving the fixed rank convex quadratic maximization in binary variables by a parallel zonotope construction algorithm. Eur. J. Oper. Res. 166, 35–50 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Gallo, G., Grigoridis, M., Tarjan, R.E.: A fast parametric maximum flow algorithm and applications. SIAM J. Comput. 18, 30–55 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. WH Freeman & Co., New York (1979)Google Scholar
  16. 16.
    Goemans, M.X., Williamson, D.P.: Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. J. Assoc. Comput. Mach. 42, 1115–1145 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Goldberg, A.V., Tarjar, R.E.: A new approach to the maximum flow problem. Proceedings of the 18th Annual ACM Symposium on Theory of Computing, Berkeley, CA, 136–146 (1986)Google Scholar
  18. 18.
    Hammer, P.L., Hansen, P., Simeone, B.: Roof duality, complementation and persistency in quadratic 0–1 optimization. Math. Program. 28, 121–155 (1984)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Hammer, P.L., Rudeanu, S.: Boolean Methods in Operations Research and Related Areas, Springer, Berlin (1968)zbMATHCrossRefGoogle Scholar
  20. 20.
    Hansen, P., Jaumard, B., Mathon, V.: Constrained nonlinear 0–1 programming. ORSA J. Comput. 5, 97–119 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Helmberg, C., Rendl, F.: Solving quadratic (0,1)-problems by semidefinite programs and cutting planes. Math. Program. 82, 291–315 (1998)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Li, D., Sun, X.L.: Nonlinear Integer Programming, Springer, New York (2006)zbMATHGoogle Scholar
  23. 23.
    Li, D., Sun, X.L., Liu, C.L.: An exact solution method for quadratic 0–1 programming: a geometric approach. Technical Report, Chinese University of Hong Kong. Department of Systems Engineering and Engineering Management (2006)Google Scholar
  24. 24.
    Mcbride, R.D., Yormark, J.S.: An implicit enumeration algorithm for quadratic integer programming. Manage. Sci. 26, 282–296 (1980)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Nemhauser, G.L., Wolsey, L.A.: Integer and Combinatorial Optimization, Wiley, New York (1988).zbMATHGoogle Scholar
  26. 26.
    Pardalos, P.M., Rodgers, G.P.: Computational aspects of a branch-and-bound algorithm for quadratic zero-one programming. Computing 45, 131–144 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Phillips, A.T., Rosen, J.B.: A quadratic assignment formulation of the molecular conformation problem. J. Global Optim. 4, 229–241 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Picard, J.C., Ratliff, H.D.: Minimum cuts and related problems. Networks 5, 357–370 (1975)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Rendl, F., Rinaldi, G., Wiegele, A.: Solving max-cut to optimality by intersecting semidefinite and polyhedral relaxations. Lect. Notes Comput. Sci. 4513, 295–309 (2007)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Rhys, J.: A selection problem of shared fixed costs and network flows. Manage. Sci. 17, 200–207 (1970)zbMATHCrossRefGoogle Scholar
  31. 31.
    Shor, N.Z.: Quadratic optimization problems. Sov. J. Comput. Syst. Sci. 25, 1–11 (1987)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Sleumer, N.: Output-sensitive cell enumeration in hyperplane arrangements. Nordic J. Comput. 6, 137–161 (1999)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Sun, X.L., Liu, C.L., Li, D., Gao, J.J.: On duality gap in binary quadratic optimization. Technical Report, Chinese University of Hong Kong. Department of Systems Engineering and Engineering Management (2007)Google Scholar
  34. 34.
    Zaslavsky, T.: Facing up to arrangements: face-count formulas for partitions of space by hyperplanes. Mem. Am. Math. Soc. 1, 1–101 (1975)MathSciNetGoogle Scholar

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© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Department of Systems Engineering and Engineering ManagementThe Chinese University of Hong KongShatinHong Kong
  2. 2.Department of Management Science, School of ManagementFudan UniversityShanghaiP. R. China
  3. 3.Department of AutomationShanghai UniversityShanghaiChina
  4. 4.Department of Applied MathematicsShanghai University of Finance and EconomicsShanghaiP. R. China

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