### 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=1}a _{ij}x _{i}x _{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 language | English (US) |
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Title of host publication | Proceedings of the 19th Annual ACM-SIAM Symposium on Discrete Algorithms |

Pages | 64-73 |

Number of pages | 10 |

State | Published - Dec 1 2008 |

Externally published | Yes |

Event | 19th Annual ACM-SIAM Symposium on Discrete Algorithms - San Francisco, CA, United States Duration: Jan 20 2008 → Jan 22 2008 |

### Publication series

Name | Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms |
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### Other

Other | 19th Annual ACM-SIAM Symposium on Discrete Algorithms |
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Country | United States |

City | San Francisco, CA |

Period | 1/20/08 → 1/22/08 |

### All Science Journal Classification (ASJC) codes

- Software
- Mathematics(all)

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## Cite this

_{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).