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

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 - 2008
Event19th Annual ACM-SIAM Symposium on Discrete Algorithms - San Francisco, CA, United States
Duration: Jan 20 2008Jan 22 2008

Other

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

Fingerprint

Hardness
Polynomials
Game
Spin glass
Approximation algorithms
Ground state
Physics
Computing
Clustering
Potential Well
Spin Glass
Complete Graph
Polynomial-time Algorithm
Ground State
Approximation Algorithms
Vanish
Polynomial time
NP-complete problem
Interpolate
Eigenvalue

ASJC Scopus subject areas

  • Software
  • Mathematics(all)

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)

The UGC hardness threshold of the l p Grothendieck problem. / Kindler, Guy; Naor, Assaf; Schechtman, Gideon.

Proceedings of the 19th Annual ACM-SIAM Symposium on Discrete Algorithms. 2008. p. 64-73.

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

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, 19th Annual ACM-SIAM Symposium on Discrete Algorithms, San Francisco, CA, United States, 1/20/08.
Kindler G, Naor A, Schechtman G. The UGC hardness threshold of the l p Grothendieck problem. In Proceedings of the 19th Annual ACM-SIAM Symposium on Discrete Algorithms. 2008. p. 64-73
Kindler, Guy ; Naor, Assaf ; Schechtman, Gideon. / The UGC hardness threshold of the l p Grothendieck problem. Proceedings of the 19th Annual ACM-SIAM Symposium on Discrete Algorithms. 2008. pp. 64-73
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