Approximate graph coloring by semidefinite programming
Journal of the ACM (JACM)
Covering a hypergraph of subgraphs
Discrete Mathematics - Kleitman and combinatorics: a celebration
FOCS '02 Proceedings of the 43rd Symposium on Foundations of Computer Science
Clustering with Qualitative Information
FOCS '03 Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science
Algorithms with large domination ratio
Journal of Algorithms
Approximating the cut-norm via Grothendieck's inequality
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
Maximizing Quadratic Programs: Extending Grothendieck's Inequality
FOCS '04 Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
Repeated communication and Ramsey graphs
IEEE Transactions on Information Theory
On Non-Approximability for Quadratic Programs
FOCS '05 Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
Correlation clustering with a fixed number of clusters
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
Towards computing the Grothendieck constant
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
Bypassing UGC from some optimal geometric inapproximability results
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
Improved approximation algorithms for MAX NAE-SAT and MAX SAT
WAOA'05 Proceedings of the Third international conference on Approximation and Online Algorithms
The Grothendieck constant of random and pseudo-random graphs
Discrete Optimization
On the complexity of Newman's community finding approach for biological and social networks
Journal of Computer and System Sciences
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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,supf:V→S|V|-1 Σ(u,v) ∈ E A(u,v) · ‹f(u),f(v)› ≤ K supf:V→(-1,+1) Σ(u,v)∈ E A(u,v) · f(u)f(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∈EA(u,v)f(vover all f: 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 θḠ)) where θ(Ḡ) 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 Θ vertices with no loops is ⏷(log n ). More generally, the maximum possible integrality gap for any perfect graph with chromatic number n is ⏷(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.