Grothendieck-Type Inequalities in Combinatorial Optimization

Communications on Pure and Applied Mathematics

Volumn 65, Issue 7, 2012, Pages 992-1035

  Khot, Subhash ^a   Naor, Assaf ^a  

^a NEW YORK UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84860273476     PISSN: 00103640     EISSN: 10970312     Source Type: Journal    
DOI: 10.1002/cpa.21398     Document Type: Article

References (117)

1. Acín, A.; Gisin, N.; Toner, B. Grothendieck's constant and local models for noisy entangled quantum states. Phys. Rev. A (3) 73 ( 2006), no. 6, part A, 062105, 5. doi:10.1103/PhysRevA.73.062105
2. n unit ball. Unpublished manuscript, 2006.
3. Alon, N.; Berger, E. The Grothendieck constant of random and pseudo-random graphs. Discrete Optim. 5 ( 2008), no. 2, 323-327 doi:10.1016/j.disopt.2006.06.004
4. Alon, N.; Coja-Oghlan, A.; Hàn, H.; Kang, M.; Rödl, V.; Schacht, M. Quasi-randomness and algorithmic regularity for graphs with general degree distributions. SIAM J. Comput. 39 ( 2010), no. 6, 2336-2362 doi:10.1137/070709529
5. Alon, N.; Duke, R. A.; Lefmann, H.; Rödl, V.; Yuster, R. The algorithmic aspects of the regularity lemma. J. Algorithms 16 ( 1994), no. 1, 80-109 doi:10.1006/jagm.1994.1005
6. Alon, N.; Fernandez de la Vega, W.; Kannan, R.; Karpinski, M. Random sampling and approximation of MAX-CSPs. J. Comput. System Sci. 67 ( 2003), no. 2, 212-243 doi:10.1016/S0022-0000(03)00008-4
7. Alon, N.; Makarychev, K.; Makarychev, Y.; Naor, A. Quadratic forms on graphs. Invent. Math. 163 ( 2006), no. 3, 499-522
8. Alon, N.; Naor, A. Approximating the cut-norm via Grothendieck's inequality. SIAM J. Comput. 35 ( 2006), no. 4, 787-803 (electronic).
9. Alon, N.; Orlitsky, A. Repeated communication and Ramsey graphs. IEEE Trans. Inform. Theory 41 ( 1995), no. 5, 1276-1289 doi:10.1109/18.412676
10. Alon, N.; Spencer, J. H. The probabilistic method. 2nd ed. Wiley-Interscience Series in Discrete Mathematics and Optimization. Wiley-Interscience, New York, 2000.
11. Arora, S.; Barak, B. Computational complexity: A modern approach. Cambridge University Press, Cambridge, 2009.
12. Arora, S.; Berger, E.; Kindler, G.; Hazan, E.; Safra, S. On non-approximability for quadratic programs. 46th Annual Symposium on Foundations of Computer Science, 206-215. IEEE Computer Society, Los Alamitos, Calif., 2005.
13. Bachas, C. P. Computer-intractability of the frustration model of a spin glass. J. Phys. A 17 ( 1984), no. 13, L709-L712. Available at:
14. Bansal, N.; Blum, A.; Chawla, S. Correlation Clustering. 43rd Annual IEEE Symposium on Foundations of Computer Science, 238-247. IEEE Computer Society, Los Alamitos, Calif., 2002.
15. Bansal, N.; Bravyi, S.; Terhal, B. M. Classical approximation schemes for the ground-state energy of quantum and classical Ising spin Hamiltonians on planar graphs. Quantum Inf. Comput. 9 ( 2009), no. 7-8, 701-720.
16. Bansal, N.; Williams, R. Regularity lemmas and combinatorial algorithms. 2009 50th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2009), 745-754. IEEE Computer Society, Los Alamitos, Calif., 2009. doi:10.1109/FOCS.2009.76
17. Barahona, F. On the computational complexity of Ising spin glass models. J. Phys. A 15 ( 1982), no. 10, 3241-3253 Available at:
18. Bhaskara, A.; Vijayaraghavan, A. Approximating matrix p-norms. Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms, 497-511. SIAM, Philadelphia, 2011.
19. Bieche, I.; Maynard, R.; Rammal, R.; Uhry, J.-P. On the ground states of the frustration model of a spin glass by a matching method of graph theory. J. Phys. A 13 ( 1980), no. 8, 2553-2576 Available at:
20. Blei, R. Analysis in integer and fractional dimensions. Cambridge Studies in Advanced Mathematics, 71. Cambridge University Press, Cambridge, 2001. doi:10.1017/CBO9780511543012
21. Blei, R. The Grothendieck inequality revisited. Preprint, 2011. arXiv:1111.7304v1 [math.FA]
22. Blei, R. C. Multidimensional extensions of the Grothendieck inequality and applications. Ark. Mat. 17 ( 1979), no. 1, 51-68 doi:10.1007/BF02385457
23. Braverman, M.; Makarychev, K.; Makarychev, Y.; Naor, A. The Grothendieck constant is strictly smaller than Krivine's bound. Preprint, 2011. arXiv:1103.6161v3 [math.FA]
24. Briët, J.; Buhrman, H.; Toner, B. A generalized Grothendieck inequality and entanglement in XOR games. Preprint, 2009. arXiv:0901.2009v1 [quant-ph]
25. Briët, J.; de Oliveira Filho, F. M.; Vallentin, F. Grothendieck inequalities for semidefinite programs with rank constraint. Preprint, 2010. arXiv:1011.1754v1 [math.OC]
26. Briët, J.; de Oliveira Filho, F. M.; Vallentin, F. The positive semidefinite Grothendieck problem with rank constraint. Automata, Languages and Programming, 37th International Colloquium, ICALP 2010, Bordeaux, France, July 6-10, 2010, Proceedings, Part I, 31-42. Lecture Notes in Computer Science, 6198. Springer, Berlin-Heidelberg-New York, 2010.
27. Charikar, M.; Guruswami, V.; Wirth, A. Clustering with qualitative information. J. Comput. System Sci. 71 ( 2005), no. 3, 360-383 doi:10.1016/j.jcss.2004.10.012
28. Charikar, M.; Makarychev, K.; Makarychev, Y On the advantage over random for maximum acyclic subgraph. 48th Annual IEEE Symposium on Foundations of Computer Science, 625-633. IEEE Computer Society, Los Alamitos, Calif., 2007.
29. Charikar, M.; Wirth, A. Maximizing quadratic programs: extending Grothendieck's inequality. 45th Annual Symposium on Foundations of Computer Science, 54-60. IEEE Computer Society, Los Alamitos, Calif., 2004.
30. Cleve, R.; Høyer, P.; Toner, B.; Watrous, J. Consequences and limits of nonlocal strategies. 19th Annual IEEE Conference on Computational Complexity, 236-249. IEEE Computer Society, Los Alamitos, Calif., 2004.
31. Coja-Oghlan, A.; Cooper, C.; Frieze, A. An efficient sparse regularity concept. SIAM J. Discrete Math. 23 ( 2009/10), no 4, 2000-2034. doi:10.1137/080730160
32. Conlon, D.; Fox, J. Bounds for graph regularity and removal lemmas. Preprint, 2011. arXiv:1107.4829v1 [math.CO]
33. Cook, S. The P versus NP problem. The millennium prize problems, 87-104. Clay Mathematics Institute, Cambridge, Mass., 2006.
34. Davie, A. M. Matrix norms related to Grothendieck's inequality. Banach spaces (Columbia, Mo., 1984), 22-26. Lecture Notes in Mathematics, 1166. Springer, Berlin, 1985. doi:10.1007/BFb0074689
35. Diestel, J.; Fourie, J. H.; Swart, J. The metric theory of tensor products: Grothendieck's résumé revisited. American Mathematical Society, Providence, R.I., 2008.
36. Diestel, J.; Jarchow, H.; Tonge, A. Absolutely summing operators. Cambridge Studies in Advanced Mathematics, 43. Cambridge University Press, Cambridge, 1995. doi:10.1017/CBO9780511526138
37. Downey, R. G.; Fellows, M. R. Parameterized complexity. Monographs in Computer Science. Springer, New York, 1999.
38. Erdos, P.; Rényi, A. On the evolution of random graphs. Magyar Tud. Akad. Mat. Kutató Int. Közl. 5 ( 1960), 17-61.
39. Fishburn, P. C.; Reeds, J. A. Bell inequalities, Grothendieck's constant, and root two. SIAM J. Discrete Math. 7 ( 1994), no. 1, 48-56 doi:10.1137/S0895480191219350
40. Frieze, A.; Kannan, R. Quick approximation to matrices and applications. Combinatorica 19 ( 1999), no. 2, 175-220 doi:10.1007/s004930050052
41. Garey, M. R.; Johnson, D. S. Computers and intractability: A guide to the theory of NP-completeness. W. H. Freeman, San Francisco, 1979.
42. Garling, D. J. H. Inequalities: a journey into linear analysis. Cambridge University Press, Cambridge, 2007. doi:10.1017/CBO9780511755217
43. Gerke, S.; Steger, A. A characterization for sparse e-regular pairs. Electron. J. Combin. 14 ( 2007), no. 1, Research Paper 4, 12 pp. (electronic). Available at:
44. Goemans, M. X.; Williamson, D. P. Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. J. Assoc. Comput. Mach. 42 ( 1995), no. 6, 1115-1145 doi:10.1145/227683.227684
45. Golub, G. H.; Van Loan, C. F. Matrix computations. 3rd ed. Johns Hopkins Studies in the Mathematical Sciences. Johns Hopkins University Press, Baltimore, Md., 1996.
46. Gowers, W. T. Lower bounds of tower type for Szemerédi's uniformity lemma. Geom. Funct. Anal. 7 ( 1997), no. 2, 322-337 doi:10.1007/PL00001621
47. Grothendieck, A. Résumé de la théorie métrique des produits tensoriels topologiques. Bol. Soc. Mat. São Paulo 8 ( 1953), 1-79.
48. Grötschel, M.; Lovász, L.; Schrijver, A. Geometric algorithms and combinatorial optimization: Algorithms and combinatorics, vol. 2. 2nd ed. Springer, Berlin, 1993.
49. Guruswami, V.; Manokaran, R.; Raghavendra, P. Beating the random ordering is hard: Inapproximability of maximum acyclic subgraph. 49th Annual IEEE Symposium on Foundations of Computer Science, 573-582. IEEE Computer Society, Los Alamitos, Calif., 2008.
50. Guruswami, V.; Raghavendra, P.; Saket, R.; Wu, Y. Bypassing UGC from some optimal geometric inapproximability results. Electronic Colloquium on Computational Complexity (ECCC) 17 ( 2010), 177.
51. Haagerup, U. The Grothendieck inequality for bilinear forms on C*-algebras. Adv. in Math. 56 ( 1985), no. 2, 93-116 doi:10.1016/0001-8708(85)90026-X
52. Haagerup, U. A new upper bound for the complex Grothendieck constant. Israel J. Math. 60 ( 1987), no. 2, 199-224
53. Håstad, J. Some optimal inapproximability results. J. ACM 48 ( 2001), no. 4, 798-859 (electronic).
54. Håstad, J.; Venkatesh, S. On the advantage over a random assignment. Random Structures Algorithms 25 ( 2004), no. 2, 117-149 doi:10.1002/rsa.20031
55. 3. Unpublished manuscript, 2011.
56. Heydari, H. Quantum correlation and Grothendieck's constant. J. Phys. A 39 ( 2006), no. 38, 11 869-11 875. doi: 10.1088/0305-4470/39/38/012
57. Jameson, G. J. O. Summing and nuclear norms in Banach space theory. London Mathematical Society Student Texts, 8. Cambridge University Press, Cambridge, 1987. doi:10.1017/CBO9780511569166
58. Johnson, W. B.; Lindenstrauss, J. Basic concepts in the geometry of Banach spaces. Handbook of the geometry of Banach spaces, vol. I, 1-84. North-Holland, Amsterdam, 2001. doi:10.1016/S1874-5849(01)80003-6
59. Judin, D. B.; Nemirovskiǐ, A. S. Informational complexity and effective methods for the solution of convex extremal problems. Èkonom. i Mat. Metody 12 ( 1976), no. 2, 357-369
60. Juhász, F. The asymptotic behaviour of Lovász' 8 function for random graphs. Combinatorica 2 ( 1982), no. 2, 153-155 doi:10.1007/BF02579314
61. Karger, D.; Motwani, R.; Sudan, M. Approximate graph coloring by semidefinite programming. J. ACM 45 ( 1998), no. 2, 246-265 doi:10.1145/274787.274791
62. Kashin, B. S.; Sharek, S. İ. On the Gram matrices of systems of uniformly bounded functions. Tr. Mat. Inst. Steklova 243 ( 2003), Funkts. Prostran., Priblizh., Differ. Uravn., 237-243.
63. Kempe, J.; Kobayashi, H.; Matsumoto, K.; Toner, B.; Vidick, T. Entangled games are hard to approximate. 49th Annual IEEE Symposium on Foundations of Computer Science, 447-456. IEEE Computer Society, Los Alamitos, Calif., 2008.
64. Khot, S. On the power of unique 2-prover 1-round games. Proceedings of the Thirty-Fourth Annual ACM Symposium on Theory of Computing, 767-775. ACM, New York, 2002 (electronic).
65. Khot, S. On the unique games conjecture (invited survey). Proceedings of the 25th Annual IEEE Conference on Computational Complexity, 99-121. IEEE Computer Society, Los Alamitos, Calif., 2010.
66. Khot, S.; Kindler, G.; Mossel, E.; O'Donnell, R. Optimal inapproximability results for MAX-CUT and other 2-variable CSPs? SIAM J. Comput. 37 ( 2007), no. 1, 319-357 (electronic).
67. Khot, S.; Naor, A. Linear equations modulo 2 and the L1 diameter of convex bodies. SIAM J. Comput. 38 ( 2008), no. 4, 1448-1463 doi:10.1137/070691140
68. Khot, S.; Naor, A. Approximate kernel clustering. Mathematika 55 ( 2009), no. 1-2, 129-165. doi: 10.1112/S002557930000098X
69. Khot, S.; Naor, A. Sharp kernel clustering algorithms and their associated Grothendieck inequalities. Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms, 664-683. SIAM, Philadelphia, 2011.
70. Khot, S.; O'Donnell, R. SDP gaps and UGC-hardness for max-cut-gain. Theory Comput. 5 ( 2009), 83-117.
71. Khot, S.; Safra, S. A two prover one round game with strong soundness. In 52ndAnnual IEEE Symposium on Foundations of Computer Science, in press.
72. p Grothendieck problem. Math. Oper. Res. 35 ( 2010), no. 2, 267-283 doi: 10.1287/moor. 1090.0425
73. Kohayakawa, Y. Szemerédi's regularity lemma for sparse graphs. Foundations of computational mathematics (Rio de Janeiro, 1997), 216-230. Springer, Berlin, 1997.
74. Kohayakawa, Y.; Rödl, V. Szemerédi's regularity lemma and quasi-randomness. Recent advances in algorithms and combinatorics, 289-351. CMS Books in Mathematics, 11. Springer, New York, 2003. doi:10.1007/0-387-22444-0_9
75. Kohayakawa, Y.; Rödl, V.; Thoma, L. An optimal algorithm for checking regularity. SIAM J. Comput. 32 ( 2003), no. 5, 1210-1235 (electronic). doi: 10.1137/S0097539702408223
76. König, H. On the complex Grothendieck constant in the n-dimensional case. Geometry of Banach spaces (Strobl, 1989), 181-198. London Mathematical Society Lecture Note Series, 158. Cambridge University Press, Cambridge, 1990.
77. König, H. Some remarks on the Grothendieck inequality. General inequalities, 6 (Oberwol-fach, 1990), 201-206. International Series of Numerical Mathematics, 103. Birkhäuser, Basel, 1992.
78. König, H. On an extremal problem originating in questions of unconditional convergence. Recent progress in multivariate approximation (Witten-Bommerholz, 2000), 185-192. International Series of Numerical Mathematics, 137. Birkhäuser, Basel, 2001.
79. Krivine, J.-L. Sur la constante de Grothendieck. C. R. Acad. Sci. Paris Sen A-B 284 ( 1977), no. 8, A445-A446.
80. Krivine, J.-L. Constantes de Grothendieck et fonctions de type positif sur les sphères. Adv. in Math. 31 ( 1979), no. 1, 16-30 doi:10.1016/0001-8708(79)90017-3
81. Laurent, M.; Varvitsiotis, A. Computing the Grothendieck constant of some graph classes. Preprint, 2011. arXiv:1106.2735v1 [math.CO]
82. Lee, T.; Schechtman, G.; Shraibman, A. Lower bounds on quantum multiparty communication complexity. Proceedings of the 24th Annual IEEE Conference on Computational Complexity, 254-262. IEEE Computer Society, Los Alamitos, Calif., 2009.
83. Lee, T.; Shraibman, A. Lower bounds in communication complexity. Found. Trends Theor. Comput. Sci. 3 ( 2007), no. 4, front matter, 263-399 (2009). doi:10.1561/0400000040
84. Lewis, A. S.; Overton, M. L. Eigenvalue optimization. Acta numerica, 1996, 149-190. Acta Numerica, 5. Cambridge University Press, Cambridge, 1996.
85. p-spaces and their applications. StudiaMath. 29 ( 1968), 275-326.
86. Linial, N.; Mendelson, S.; Schechtman, G.; Shraibman, A. Complexity measures of sign matrices. Combinatorica 27 ( 2007), no. 4, 439-463 doi:10.1007/s00493-007-2160-5
87. Linial, N.; Shraibman, A. Learning complexity vs. communication complexity. Combin. Probab. Comput. 18 ( 2009), no. 1-2, 227-245. doi:10.1017/S0963548308009656
88. Linial, N.; Shraibman, A. Lower bounds in communication complexity based on factorization norms. Random Structures Algorithms 34 ( 2009), no. 3, 368-394 doi:10.1002/rsa.20232
89. Lovász, L. On the Shannon capacity of a graph. IEEE Trans. Inform. Theory 25 ( 1979), no. 1, 1-7 doi: 10.1109/TIT.1979.1055985
90. Lovász, L.; Plummer, M. D. Matching theory. North-Holland Mathematics Studies, 121. Annals of Discrete Mathematics, 29. North-Holland, Amsterdam, 1986.
91. Lovász, L.; Szegedy, B. Szemerédi's lemma for the analyst. Geom. Funct. Anal. 17 ( 2007), no. 1, 252-270 doi: 10.1007/s00039-007-0599-6
92. Matula, D. W. On the complete subgraphs of a random graph. Proc. Second Chapel Hill Conf. on Combinatorial Mathematics and Its Applications (Univ. North Carolina, Chapel Hill, N.C., 1970), 356-369. University of North Carolina, Chapel Hill, N.C., 1970.
93. Megretski, A. Relaxations of quadratic programs in operator theory and system analysis. Systems, approximation, singular integral operators, and related topics (Bordeaux, 2000), 365-392. Operator Theory: Advances and Applications, 129. Birkhäuser, Basel, 2001.
94. Naor, A.; Schechtman, G. An approximation scheme for quadratic form maximization on convex bodies. Unpublished manuscript, 2009.
95. Nemirovski, A. Advances in convex optimization: Conic programming. International Congress of Mathematicians, Vol. I, 413-444. European Mathematical Society, Zürich, 2007. doi:10.4171/022-1/17
96. Nemirovski, A.; Roos, C.; Terlaky, T. On maximization of quadratic form over intersection of ellipsoids with common center. Math. Program. 86 ( 1999), no. 3, Ser. A, 463-473.
97. Nesterov, Y. Semidefinite relaxation and nonconvex quadratic optimization. Optim. Methods Softw. 9 ( 1998), no. 1-3, 141-160. doi:10.1080/10556789808805690
98. Nesterov, Y.; Wolkowicz, H.; Ye, Y Semidefinite programming relaxations of nonconvex quadratic optimization. Handbook of semidefinite programming<, 361-419. International Series in Operations Research & Management Science, 27. Kluwer, Boston, 2000.
99. Papadimitriou, C. H.; Yannakakis, M. Optimization, approximation, and complexity classes. J. Comput. System Sci. 43 ( 1991), no. 3, 425-440 doi:10.1016/0022-0000(91)90023-X
100. Pérez-García, D.; Wolf, M. M.; Palazuelos, C.; Villanueva, I.; Junge, M. Unbounded violation of tripartite Bell inequalities. Comm. Math. Phys. 279 ( 2008), no. 2, 455-186. doi: 10.1007/s00220-008-0418-4
101. Pisier, G. Grothendieck's theorem for noncommutative C*-algebras, with an appendix on Grothendieck's constants. J. Funct. Anal. 29 ( 1978), no. 3, 397-415
102. Pisier, G. Factorization of linear operators and geometry ofBanach spaces. CBMS Regional Conference Series in Mathematics, 60. American Mathematical Society, Providence, R.I., 1986.
103. Pisier, G. Grothendieck's theorem, past and present. Preprint, 2011. arXiv:1101.4195v3 [math.FA]
104. Pitowsky, I. New Bell inequalities for the singlet state: going beyond the Grothendieck bound. J. Math. Phys. 49 ( 2008), no. 1, 012 101, 11. doi: 10.1063/1.2826227
105. Raghavendra, P. Optimal algorithms and inapproximability results for every CSP? Proceedings of the 40th Annual ACM Symposium on Theory of Computing, 245-254. ACM, New York, 2008.
106. Raghavendra, P.; Steurer, D. Towards computing the Grothendieck constant. Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms, 525-534. SIAM, Philadelphia, 2008.
107. Reeds, J. A. A new lower bound on the real Grothendieck constant. Unpublished manuscript, 1991. Available at:
108. Regev, O.; Toner, B. Simulating quantum correlations with finite communication. SIAM J. Comput. 39 ( 2009/10), no. 4, 1562-1580. doi: 10.1137/080723909
109. Rietz, R. E. A proof of the Grothendieck inequality. Israel J. Math. 19 ( 1974), 271-276.
110. Sipser, M. Introduction to the theory of computation. PWS Publishing, Boston, 1997.
111. Smith, R. R. Completely bounded multilinear maps and Grothendieck's inequality. Bull. London Math. Soc. 20 ( 1988), no. 6, 606-612 doi:10.1112/blms/20.6.606
112. Song, L.; Smola, A.; Gretton, A.; Borgwardt, K. A. A dependence maximization view of clustering. Proceedings of the 24th International Conference on Machine Learning, 815-822. ACM, New York, 2007. Available at:
113. Szemerédi, E. Regular partitions of graphs. Problèmes combinatoires et théorie des graphes (Colloq. Internat. CNRS, Univ. Orsay, Orsay, 1976), 399-401. Colloques Internationaux du Centre National de la Recherche Scientifique, 260. CNRS, Paris, 1978.
114. Tonge, A. The von Neumann inequality for polynomials in several Hilbert-Schmidt operators. J. LondonMath. Soc. (2) 18 ( 1978), no. 3, 519-526 doi:10.1112/jlms/s2-18.3.519
115. Tonge, A. The complex Grothendieck inequality for 2 × 2 matrices. Bull. Soc. Math. Grèce (N.S.) 27 ( 1986), 133-136.
116. Tsirelson, B. S. Quantum analogues of Bell's inequalities. The case of two spatially divided domains. Problems of the theory of probability distributions, IX. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 142 ( 1985), 174-194, 200.
117. Varopoulos, N. T. On an inequality of von Neumann and an application of the metric theory of tensor products to operators theory. J. Functional Analysis 16 ( 1974), 83-100.