Article

Inventiones mathematicae

, Volume 163, Issue 3, pp 499-522

First online:

Quadratic forms on graphs

  • Noga AlonAffiliated withSchools of Mathematics and Computer Science, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv UniversityInstitute for Advanced Study Email author 
  • , Konstantin MakarychevAffiliated withDepartment of Computer Science, Princeton University Email author 
  • , Yury MakarychevAffiliated withDepartment of Computer Science, Princeton University Email author 
  • , Assaf NaorAffiliated withMicrosoft Research Email author 

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Abstract

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.