The UGC hardness threshold of the l p Grothendieck problem

Guy Kindler, Assaf Naor, Gideon Schechtman

Research output: Chapter in Book/Report/Conference proceedingConference contribution

4 Scopus citations

Abstract

For p ≥ 2 we consider the problem of, given an n × n matrix A = (a ij) whose diagonal entries vanish, approximating in polynomial time the number Opt p(A):= max{Σ n i,j=1a ijx ix j: (Σ n i=1|x i| p) 1/P ≤1} (where optimization is taken over real numbers). When p = 2 this is simply the problem of computing the maximum eigenvalue of A, while for p = ∞ (actually it suffices to take p ≈ log n) it is the Grothendieck problem on the complete graph, which was shown to have a O(log n) approximation algorithm in[27, 26, 15], and was used in[15] to design the best known algorithm for the problem of computing the maximum correlation in Correlation Clustering. Thus the problem of approximating Opt p(A) interpolates between the spectral (p = 2) case and the Correlation Clustering (p = ∞) case. From a physics point of view this problem corresponds to computing the ground states of spin glasses in a hard-wall potential well. We design a polynomial time algorithm which, given p ≥ 2 and an n × n matrix A = (a ij) with zeros on the diagonal, computes Opt p(A) up to a factor p/e 30 log p. On the other hand, assuming the unique games conjecture (UGC) we show that it is NP-hard to approximate (1.2) up to a factor smaller than p/e+1/4. Hence as p → ∞ the UGC-hardness threshold for computing Opt p(A) is exactly p/e (1 + o(1)).

Original languageEnglish (US)
Title of host publicationProceedings of the 19th Annual ACM-SIAM Symposium on Discrete Algorithms
Pages64-73
Number of pages10
StatePublished - Dec 1 2008
Externally publishedYes
Event19th Annual ACM-SIAM Symposium on Discrete Algorithms - San Francisco, CA, United States
Duration: Jan 20 2008Jan 22 2008

Publication series

NameProceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms

Other

Other19th Annual ACM-SIAM Symposium on Discrete Algorithms
CountryUnited States
CitySan Francisco, CA
Period1/20/081/22/08

All Science Journal Classification (ASJC) codes

  • Software
  • Mathematics(all)

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  • Cite this

    Kindler, G., Naor, A., & Schechtman, G. (2008). The UGC hardness threshold of the l p Grothendieck problem. In Proceedings of the 19th Annual ACM-SIAM Symposium on Discrete Algorithms (pp. 64-73). (Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms).