### Abstract

For p ≥ 2 we consider the problem of, given an n x n matrix A = (a _{ij}) whose diagonal entries vanish, approximating in polynomial time the number Opt_{p}(A):= max { σ_{i, j=1}
^{n} a_{ij}x_{i}x_{j}: (x_{1},....,x_{n}) ∈ ℝ^{n} ∧ (σ_{i=1}
^{n}|x _{i}|^{p})^{1/p}≤1}.When p = 2 this is simply the problem of computing the maximum eigenvalue of A, whereas 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 0(log n) approximation algorithm in Nemirovski et al. [Nemirovski, A., C. Roos, T. Terlaky. 1999. On maximization of quadratic form over intersection of ellipsoids with common center. Math. Program. Ser. A 86(3) 463-473], Megretski [Megretski, A. 2001. Relaxations of quadratic programs in operator theory and system analysis. Systems, Approximation, Singular Integral Operators, and Related Topics (Bordeaux, 2000), Vol. 129. Operator Theory Advances and Applications. Birkhäuser, Basel, 365-392], Charikar and Wirth [Charikar, M., A. Wirth. 2004. Maximizing quadratic programs: Extending Grothendieck's inequality. Proc. 45th Annual Sympos. Foundations Comput. Sci., IEEE Computer Society, 54-60] and was used in the work of Charikar and Wirth noted above, 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 x n matrix A = (a_{ij}) with zeros on the diagonal, computes Opt_{p}(A) up to a factor p/e + 30logp. On the other hand, assuming the unique games conjecture (UGC) we show that it is NP-hard to approximate Opt_{p}(A) 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) |
---|---|

Pages (from-to) | 267-283 |

Number of pages | 17 |

Journal | Mathematics of Operations Research |

Volume | 35 |

Issue number | 2 |

DOIs | |

State | Published - May 2010 |

### Fingerprint

### Keywords

- Approximation algorithms
- Quadratic programming
- Unique games hardness

### ASJC Scopus subject areas

- Mathematics(all)
- Computer Science Applications
- Management Science and Operations Research

### Cite this

*Mathematics of Operations Research*,

*35*(2), 267-283. https://doi.org/10.1287/moor.1090.0425

**The UGC hardness threshold of the Lp grothendieck problem.** / Kindler, Guy; Naor, Assaf; Schechtman, Gideon.

Research output: Contribution to journal › Article

*Mathematics of Operations Research*, vol. 35, no. 2, pp. 267-283. https://doi.org/10.1287/moor.1090.0425

}

TY - JOUR

T1 - The UGC hardness threshold of the Lp grothendieck problem

AU - Kindler, Guy

AU - Naor, Assaf

AU - Schechtman, Gideon

PY - 2010/5

Y1 - 2010/5

N2 - For p ≥ 2 we consider the problem of, given an n x n matrix A = (a ij) whose diagonal entries vanish, approximating in polynomial time the number Optp(A):= max { σi, j=1 n aijxixj: (x1,....,xn) ∈ ℝn ∧ (σi=1 n|x i|p)1/p≤1}.When p = 2 this is simply the problem of computing the maximum eigenvalue of A, whereas 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 0(log n) approximation algorithm in Nemirovski et al. [Nemirovski, A., C. Roos, T. Terlaky. 1999. On maximization of quadratic form over intersection of ellipsoids with common center. Math. Program. Ser. A 86(3) 463-473], Megretski [Megretski, A. 2001. Relaxations of quadratic programs in operator theory and system analysis. Systems, Approximation, Singular Integral Operators, and Related Topics (Bordeaux, 2000), Vol. 129. Operator Theory Advances and Applications. Birkhäuser, Basel, 365-392], Charikar and Wirth [Charikar, M., A. Wirth. 2004. Maximizing quadratic programs: Extending Grothendieck's inequality. Proc. 45th Annual Sympos. Foundations Comput. Sci., IEEE Computer Society, 54-60] and was used in the work of Charikar and Wirth noted above, to design the best known algorithm for the problem of computing the maximum correlation in correlation clustering. Thus the problem of approximating Optp(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 x n matrix A = (aij) with zeros on the diagonal, computes Optp(A) up to a factor p/e + 30logp. On the other hand, assuming the unique games conjecture (UGC) we show that it is NP-hard to approximate Optp(A) up to a factor smaller than p/e + 1 4. Hence as p --∞ the UGC-hardness threshold for computing Optp(A) is exactly (p/e)(1 + o(1)).

AB - For p ≥ 2 we consider the problem of, given an n x n matrix A = (a ij) whose diagonal entries vanish, approximating in polynomial time the number Optp(A):= max { σi, j=1 n aijxixj: (x1,....,xn) ∈ ℝn ∧ (σi=1 n|x i|p)1/p≤1}.When p = 2 this is simply the problem of computing the maximum eigenvalue of A, whereas 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 0(log n) approximation algorithm in Nemirovski et al. [Nemirovski, A., C. Roos, T. Terlaky. 1999. On maximization of quadratic form over intersection of ellipsoids with common center. Math. Program. Ser. A 86(3) 463-473], Megretski [Megretski, A. 2001. Relaxations of quadratic programs in operator theory and system analysis. Systems, Approximation, Singular Integral Operators, and Related Topics (Bordeaux, 2000), Vol. 129. Operator Theory Advances and Applications. Birkhäuser, Basel, 365-392], Charikar and Wirth [Charikar, M., A. Wirth. 2004. Maximizing quadratic programs: Extending Grothendieck's inequality. Proc. 45th Annual Sympos. Foundations Comput. Sci., IEEE Computer Society, 54-60] and was used in the work of Charikar and Wirth noted above, to design the best known algorithm for the problem of computing the maximum correlation in correlation clustering. Thus the problem of approximating Optp(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 x n matrix A = (aij) with zeros on the diagonal, computes Optp(A) up to a factor p/e + 30logp. On the other hand, assuming the unique games conjecture (UGC) we show that it is NP-hard to approximate Optp(A) up to a factor smaller than p/e + 1 4. Hence as p --∞ the UGC-hardness threshold for computing Optp(A) is exactly (p/e)(1 + o(1)).

KW - Approximation algorithms

KW - Quadratic programming

KW - Unique games hardness

UR - http://www.scopus.com/inward/record.url?scp=77953100811&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77953100811&partnerID=8YFLogxK

U2 - 10.1287/moor.1090.0425

DO - 10.1287/moor.1090.0425

M3 - Article

VL - 35

SP - 267

EP - 283

JO - Mathematics of Operations Research

JF - Mathematics of Operations Research

SN - 0364-765X

IS - 2

ER -