Inventiones mathematicae

, Volume 163, Issue 3, pp 499–522 | Cite as

Quadratic forms on graphs

  • Noga Alon
  • Konstantin Makarychev
  • Yury Makarychev
  • Assaf Naor


We introduce a new graph parameter, called the Grothendieck constant of a graph G=(V,E), which is defined as the least constant K such that for every A:E→ℝ,
$$\sup_{f:V\to{S}^{|V|-1}}\sum_{\{u,v\}\in{E}} A(u,v)\cdot\langle{f(u),f(v)}\rangle\le{K}\sup_{\varphi:V\to\{-1,+1\}}\sum_{\{u,v\}\in{E}}A(u,v)\cdot\varphi(u)\varphi(v).$$
The classical Grothendieck inequality corresponds to the case of bipartite graphs, but the case of general graphs is shown to have various algorithmic applications. Indeed, our work is motivated by the algorithmic problem of maximizing the quadratic form ∑{u,v}∈E A(u,v)ϕ(u)ϕ(v) over all ϕ:V→{-1,1}, which arises in the study of correlation clustering and in the investigation of the spin glass model. We give upper and lower estimates for the integrality gap of this program. We show that the integrality gap is \(O(\log\vartheta(\overline{G}))\), where \(\vartheta(\overline{G})\) is the Lovász Theta Function of the complement of G, which is always smaller than the chromatic number of G. This yields an efficient constant factor approximation algorithm for the above maximization problem for a wide range of graphs G. We also show that the maximum possible integrality gap is always at least Ω(log ω(G)), where ω(G) is the clique number of G. In particular it follows that the maximum possible integrality gap for the complete graph on n vertices with no loops is Θ(logn). More generally, the maximum possible integrality gap for any perfect graph with chromatic number n is Θ(logn). The lower bound for the complete graph improves a result of Kashin and Szarek on Gram matrices of uniformly bounded functions, and settles a problem of Megretski and of Charikar and Wirth.


Quadratic Form Bipartite Graph Complete Graph Spin Glass Theta Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alon, N.: Explicit Ramsey graphs and orthonormal labelings. Electron. J. Comb. 1, R12 (1994)Google Scholar
  2. 2.
    Alon, N.: Covering a hypergraph of subgraphs. Discrete Math. 257, 249–254 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Alon, N., Naor, A.: Approximating the Cut-Norm via Grothendieck’s Inequality. Proc. of the 36th ACM STOC, pp. 72–80, 2004Google Scholar
  4. 4.
    Alon, N., Gutin, G., Krivelevich, M.: Algorithms with large domination ratio. J. Algorithms 50, 118–131 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Alon, N., Orlitsky, A.: Repeated communication and Ramsey graphs: IEEE Trans. Inf. Theory 41, 1276–1289 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Bansal, N., Blum, A., Chowla, S.: Correlation Clustering. Proc. of the 43 IEEE FOCS, pp. 238–247, 2002Google Scholar
  7. 7.
    Barahona, F.: On the computational complexity of Ising spin glass models. J. Phys. A, Math. Gen. 15, 3241–3253 (1982)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Bonamie, A.: Etude de coefficients Fourier des fonctiones de L p(G). Ann. Inst. Fourier 20, 335–402 (1970)Google Scholar
  9. 9.
    Charikar, M., Guruswami, V., Wirth, A.: Clustering with qualitative information. Proc. of the 44 IEEE FOCS, pp. 524–533, 2003Google Scholar
  10. 10.
    Charikar, M., Wirth, A.: Maximizing quadratic programs: extending Grothendieck’s Inequality, pp. 54–60. FOCS 2004Google Scholar
  11. 11.
    Ding, G., Seymour, P.D., Winkler, P.: Bounding the vertex cover number of a hypergraph. Combinatorica 14, 23–34 (1994)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Frieze, A.M., Kannan, R.: Quick Approximation to matrices and applications. Combinatorica 19, 175–200 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Grötschel, M., Lovász, L., Schrijver, A.: The ellipsoid method and its consequences in combinatorial optimization. Combinatorica 1, 169–197 (1981)zbMATHMathSciNetGoogle Scholar
  14. 14.
    Grothendieck, A.: Résumé de la théorie métrique des produits tensoriels topologiques. Bol. Soc. Mat. Sao Paolo 8, 1–79 (1953)MathSciNetGoogle Scholar
  15. 15.
    Johnson, W.B., Lindenstrauss, J.: Basic concepts in the geometry of Banach spaces. Handbook of the geometry of Banach spaces, vol. I, pp. 1–84. Amsterdam: North-Holland 2001Google Scholar
  16. 16.
    Juhász, F.: The asymptotic behaviour of Lovász’ θ function for random graphs. Combinatorica 2, 153–155 (1982)zbMATHMathSciNetGoogle Scholar
  17. 17.
    Kim, S.-J., Kostochka, A., Nakprasit, K.: On the Chromatic Number of Intersection Graphs of Convex Sets in the Plane. Electron. J. Comb. 11, R52 (2004)Google Scholar
  18. 18.
    Karger, D., Motwani, R., Sudan, M.: Approximate graph coloring by semidefinite programming. J. ACM 45, 246–265 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Kashin, B.S., Szarek, S.J.: On the Gram Matrices of Systems of Uniformly Bounded Functions. Proc. Steklov Inst. Math., vol. 243, pp. 227–233, 2003Google Scholar
  20. 20.
    Krivine, J.: Sur la constante de Grothendieck. C. R. Acad. Sci., Paris, Sér. A-B 284, 445–446 (1977)zbMATHMathSciNetGoogle Scholar
  21. 21.
    Lindenstrauss, J., Pełczyński, A.: Absolutely summing operators in L p spaces and their applications. Studia Math. 29, 275–326 (1968)zbMATHMathSciNetGoogle Scholar
  22. 22.
    Lovász, L.: Kneser’s conjecture, chromatic number and homotopy, J. Comb. Theory 25, 319–324 (1978)CrossRefzbMATHGoogle Scholar
  23. 23.
    Lovász, L.: On the Shannon capacity of a graph. IEEE Trans. Inf. Theory 25, 1–7 (1979)CrossRefzbMATHGoogle Scholar
  24. 24.
    Lovász, L., Plummer, M.D.: Matching Theory. Amsterdam: North Holland 1986Google Scholar
  25. 25.
    Megretski, A.: Relaxation of Quadratic Programs in Operator Theory and System Analysis. In: Systems, Approximation, Singular Integral Operators, and Related Topics (Bordeaux, 2000), pp. 365–392. Basel: Birkhäuser 2001Google Scholar
  26. 26.
    Nemirovski, A., Roos, C., Terlaky, T.: On Maximization of Quadratic Form over Intersection of Ellipsoids with Common Center. Math. Program. 86, 463–473 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Talagrand, M.: Spin glasses: a challenge for mathematicians. Cavity and mean field models, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, 46. Berlin: Springer 2003Google Scholar

Copyright information

© Springer-Verlag 2005

Authors and Affiliations

  1. 1.Schools of Mathematics and Computer Science, Raymond and Beverly Sackler Faculty of Exact SciencesTel Aviv UniversityTel AvivIsrael
  2. 2.Institute for Advanced StudyPrincetonUSA
  3. 3.Department of Computer SciencePrinceton UniversityPrincetonUSA
  4. 4.Microsoft ResearchRedmondUSA

Personalised recommendations