Quadratic forms on graphs

• 成果类型: Article
• 署名作者: Alon, N; Makarychev, K; Makarychev, Y; Naor, A
• 署名单位: Tel Aviv University; Tel Aviv University; Institute for Advanced Study - USA; Princeton University; Microsoft
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-005-0465-9
• 发表日期: 2006
• 页码: 499-522
• 关键词: chromatic number; intersection; hypergraph

摘要

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 -->S\V\-1{u, v} is an element of E) Sigma A(u, v) . [f(u), f(v)] <= K sup(phi:V --> {-1, +1} {u, v} is an element of E) Sigma A(u, v) . phi*u)phi(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 Sigma({u, v} is an element of E) A(u, v)phi(u)phi(v) over all phi : 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 theta((G) over bar)), where theta((G) over bar) is the Lovasz 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 Omega(log omega(G)), where omega(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 Theta(log n). More generally, the maximum possible integrality gap for any perfect graph with chromatic number n is Theta(log n). 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.