Journal of Global Optimization

, Volume 60, Issue 3, pp 393–423 | Cite as

Handelman’s hierarchy for the maximum stable set problem

  • Monique Laurent
  • Zhao SunEmail author


The maximum stable set problem is a well-known NP-hard problem in combinatorial optimization, which can be formulated as the maximization of a quadratic square-free polynomial over the (Boolean) hypercube. We investigate a hierarchy of linear programming relaxations for this problem, based on a result of Handelman showing that a positive polynomial over a polytope with non-empty interior can be represented as conic combination of products of the linear constraints defining the polytope. We relate the rank of Handelman’s hierarchy with structural properties of graphs. In particular we show a relation to fractional clique covers which we use to upper bound the Handelman rank for perfect graphs and determine its exact value in the vertex-transitive case. Moreover we show two upper bounds on the Handelman rank in terms of the (fractional) stability number of the graph and compute the Handelman rank for several classes of graphs including odd cycles and wheels and their complements. We also point out links to several other linear and semidefinite programming hierarchies.


Polynomial optimization Combinatorial optimization  Handelman hierarchy Linear programming relaxation The maximum stable set problem 



We thank E. De Klerk and J.C. Vera for useful discussions. We also thank two anonymous referees for their comments which helped improve the clarity of the paper and for drawing our attention to the paper by Krivine [15].


  1. 1.
    Handelman, D.: Representing polynomials by positive linear functions on compact convex polyhedra. Pac. J. Math. 132(1), 35–62 (1988)CrossRefGoogle Scholar
  2. 2.
    De Klerk, E., Laurent, M.: Error bounds for some semidefinite programming approaches to polynomial optimization on the hypercube. SIAM J. Optim. 20(6), 3104–3120 (2010)CrossRefGoogle Scholar
  3. 3.
    Park, M.-J., Hong, S.-P.: Rank of Handelman hierarchy for max-cut. Oper. Res. Lett. 39(5), 323–328 (2011)CrossRefGoogle Scholar
  4. 4.
    Park, M.-J., Hong, S.-P.: Handelman rank of zero-diagonal quadratic programs over a hypercube and its applications. J. Glob. Optim. (2012). doi: 10.1007/s10898-012-9906-3 Google Scholar
  5. 5.
    Bruck, J., Blaum, M.: Neural networks, error-correcting codes, and polynomials over the binary \(n\)-cube. IEEE Trans. Inf. Theory 35(5), 976–987 (1989)CrossRefGoogle Scholar
  6. 6.
    Cornaz, D., Jost, V.: A one-to-one correspondance between colorings and stable sets. Oper. Res. Lett. 36(6), 673–676 (2008)CrossRefGoogle Scholar
  7. 7.
    Lovász, L.: Stable sets and polynomials. Discret Math. 124(1–3), 137–153 (1994)CrossRefGoogle Scholar
  8. 8.
    De Loera, J., Lee, J., Margulies, S., Onn, S.: Expressing combinatorial problems by systems of polynomial equations and Hilbert’s Nullstellensatz. Comb. Probab. Comput. 18(4), 551–582 (2009)CrossRefGoogle Scholar
  9. 9.
    De Klerk, E., Pasechnik, D.V.: Approximating of the stability number of a graph via copositive programming. SIAM J. Optim. 12(4), 875–892 (2002)CrossRefGoogle Scholar
  10. 10.
    Gouveia, J., Parrilo, P., Thomas, R.: Theta bodies for polynomial ideals. SIAM J. Optim. 20(4), 2097–2118 (2010)CrossRefGoogle Scholar
  11. 11.
    Lasserre, J.B.: An explicit equivalent positive semidefinite program for nonlinear 0–1 programs. SIAM J. Optim. 12, 756–769 (2002)CrossRefGoogle Scholar
  12. 12.
    Laurent, M.: A comparison of the Sherali–Adams, Lovász–Schrijver and Lasserre relaxation for 0–1 programming. Math. Oper. Res. 28(3), 470–498 (2003)CrossRefGoogle Scholar
  13. 13.
    Peña, J.C., Vera, J.C., Zuluaga, L.F.: Computing the stability number of a graph via linear and semidefinite programming. SIAM J. Optim. 18(1), 87–105 (2007)CrossRefGoogle Scholar
  14. 14.
    Sherali, H.D., Adams, W.P.: A hierarchy of relaxations between the continuous and convex hull representations for 0–1 programming problems. SIAM J. Discret Math. 3(3), 411–430 (1990)CrossRefGoogle Scholar
  15. 15.
    Krivine, J.L.: Quelques propriétés des préordres dans les anneaux commutatifs unitaires. C. R. Acad. Sci. Paris 258, 3417–3418 (1964)Google Scholar
  16. 16.
    Laurent, M.: Sums of squares, moment matrices and optimization over polynomials. In: Putinar, M., Sullivant, S. (eds.) Emerging Applications of Algebraic Geometry, pp. 157–270. Springer, New York (2009)CrossRefGoogle Scholar
  17. 17.
    Faybusovich, L.: Global optimization of homogeneous polynomials on the simplex and on the sphere. In: Floudas, C., Pardalos, P. (eds.) Frontiers in Global Optimization, pp. 109–121. Kluwer Academic Publishers, Dordrecht (2003)Google Scholar
  18. 18.
    De Klerk, E., Laurent, M., Parrilo, P.: A PTAS for the minimization of polynomials of fixed degree over the simplex. Theor. Comput. Sci. 361(2–3), 210–225 (2006)CrossRefGoogle Scholar
  19. 19.
    De Klerk, E., Laurent, M., Parrilo, P.: On the equivalence of algebraic approaches to the minimization of forms on the simplex. In: Henrion, D., Garulli, A. (eds.) Positive Polynomials in Control, pp. 121–133. Springer, Berlin (2005)Google Scholar
  20. 20.
    Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W. (eds.) Complexity of Computer Computations, pp. 85–103. Springer, New York (1972)CrossRefGoogle Scholar
  21. 21.
    Nemhauser, G.L., Trotter Jr, L.E.: Properties of vertex packing and independence system polyhedra. Math. Program 6(1), 48–61 (1974)CrossRefGoogle Scholar
  22. 22.
    Lovász, L., Schrijver, A.: Cones of matrices and set-functions and 0–1 optimization. SIAM J. Optim. 1(2), 166–190 (1991)CrossRefGoogle Scholar
  23. 23.
    Lovász, L.: A characterization of perfect graphs. J. Comb. Theory B. 13(2), 95–98 (1972)CrossRefGoogle Scholar
  24. 24.
    Schrijver, A.: Combinatorial Optimization: Polyhedra and Efficiency. Springer, Berlin (2003)Google Scholar
  25. 25.
    Chudnovsky, M., Robertson, N., Seymour, P., Thomas, R.: The strong perfect graph theorem. Ann. Math. 164(1), 51–229 (2006)CrossRefGoogle Scholar
  26. 26.
    Lipták, L., Tuncel, L.: The stable set problem and the lift-and-project ranks of graphs. Math. Program B 98(1–3), 319–353 (2003)CrossRefGoogle Scholar
  27. 27.
    Lasserre, J.B.: Semidefinite programming vs. LP relaxations for polynomial programming. Math. Oper. Res. 27(2), 347–360 (2002)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Centrum Wiskunde & Informatica (CWI)AmsterdamThe Netherlands
  2. 2.Tilburg UniversityTilburgThe Netherlands

Personalised recommendations