Explicit construction of linear sized tolerant networks
Discrete Mathematics - First Japan Conference on Graph Theory and Applications
Maximizing Quadratic Programs: Extending Grothendieck's Inequality
FOCS '04 Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
The UGC hardness threshold of the ℓp Grothendieck problem
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
The UGC Hardness Threshold of the Lp Grothendieck Problem
Mathematics of Operations Research
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The Grothendieck constant of a graph G=(V,E) is the least constant K such that for every matrix A:VxV-R, maxf:V-S^|^V^|^-^1@?{u,v}@?EA(u,v)@?@?Kmax@e:V-{-1,+1}@?{u,v}@?EA(u,v)@?@e(u)@e(v). The investigation of this parameter, introduced in [N. Alon, K. Makarychev, Y. Makarychev, A. Naor, Quadratic forms on graphs, in: Proc. of the 37th ACM STOC, ACM Press, Baltimore, 2005, pp. 486-493 (Also: Invent. Math. 163 (2006) 499-522)], is motivated by the algorithmic problem of maximizing the quadratic form @?"{"u","v"}"@?"EA(u,v)@e(u)@e(v) over all @e:V-{-1,1}, which arises in the study of correlation clustering and in the investigation of the spin glass model. In the present note we show that for the random graph G(n,p) the value of this parameter is, almost surely, @Q(log(np)). This settles a problem raised in [N. Alon, K. Makarychev, Y. Makarychev, A. Naor, Quadratic forms on graphs, in: Proc. of the 37th ACM STOC, ACM Press, Baltimore, 2005, pp. 486-493 (Also: Invent. Math. 163 (2006) 499-522)]. We also obtain a similar estimate for regular graphs in which the absolute value of each nontrivial eigenvalue is small.