TY - GEN

T1 - On non-approximability for quadratic programs

AU - Arora, Sanjeev

AU - Berger, Eli

AU - Hazan, Elad

AU - Kindler, Guy

AU - Safra, Muli

PY - 2005

Y1 - 2005

N2 - This paper studies the computational complexity of the following type of quadratic programs: given an arbitrary matrix whose diagonal elements are zero, find x ε {-1,1} n that maximizes x TMx. This problem recently attracted attention due to its application in various clustering settings, as well as an intriguing connection to the famous Grothendieck inequality. It is approximate to within a factor of O(log n), and known to be NP-hard to approximate within any factor better than 13/11 - ε for all ε > 0. We show that it is quasi-NP-hard to approximate to a factor better than O(log γ n)for some γ > 0. The integrality gap of the natural semidefinite relaxation for this problem is known as the Grothendieck constant of the complete graph, and known to be ⊖(log n). The proof of this fact was nonconstructive, and did not yield an explicit problem instance where this integrality gap is achieved. Our techniques yield an explicit instance for which the integrality gap is Ω(log n/log log n), essentially answering one of the open problems of Alon et al. [AMMN].

AB - This paper studies the computational complexity of the following type of quadratic programs: given an arbitrary matrix whose diagonal elements are zero, find x ε {-1,1} n that maximizes x TMx. This problem recently attracted attention due to its application in various clustering settings, as well as an intriguing connection to the famous Grothendieck inequality. It is approximate to within a factor of O(log n), and known to be NP-hard to approximate within any factor better than 13/11 - ε for all ε > 0. We show that it is quasi-NP-hard to approximate to a factor better than O(log γ n)for some γ > 0. The integrality gap of the natural semidefinite relaxation for this problem is known as the Grothendieck constant of the complete graph, and known to be ⊖(log n). The proof of this fact was nonconstructive, and did not yield an explicit problem instance where this integrality gap is achieved. Our techniques yield an explicit instance for which the integrality gap is Ω(log n/log log n), essentially answering one of the open problems of Alon et al. [AMMN].

UR - http://www.scopus.com/inward/record.url?scp=33748614322&partnerID=8YFLogxK

U2 - 10.1109/SFCS.2005.57

DO - 10.1109/SFCS.2005.57

M3 - Conference contribution

AN - SCOPUS:33748614322

SN - 0769524680

SN - 9780769524689

T3 - Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS

SP - 206

EP - 215

BT - Proceedings - 46th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2005

T2 - 46th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2005

Y2 - 23 October 2005 through 25 October 2005

ER -