This paper studies the computational complexity of the following type of quadratic programs: given an arbitrary matrix whose diagonal elements are zero, find x /spl isin/ {-1, 1}/sup n/ that maximizes x/sup T/Mx. 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 approximable to within a factor of O(log n), and known to be NP-hard to approximate within any factor better than 13/11 - /spl epsi/ for all /spl epsi/ > 0. We show that it is quasi-NP-hard to approximate to a factor better than O(log/sup /spl gamma// n)for some /spl gamma/ > 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 /spl Theta/(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 /spl Omega/ (log n/log log n), essentially answering one of the open problems of Alon et al. [AMMN].
This paper studies the computational complexity of the following type of quadratic programs: given an arbitrary matrix whose diagonal elements are zero, find x /spl isin/ {-1, 1}/sup n/ that maximizes x/sup T/Mx. 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 approximable to within a factor of O(log n), and known to be NP-hard to approximate within any factor better than 13/11 - /spl epsi/ for all /spl epsi/ > 0. We show that it is quasi-NP-hard to approximate to a factor better than O(log/sup /spl gamma// n)for some /spl gamma/ > 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 /spl Theta/(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 /spl Omega/ (log n/log log n), essentially answering one of the open problems of Alon et al. [AMMN].
This paper considers the following type of quadratic programming problem. Given an arbitrary matrix A, whose diagonal elements are zero, find x /spl isin/ {-1, 1}/sup n/ such that x/sup T/Ax is maximized. Our approximation algorithm for this problem uses the canonical semidefinite relaxation and returns a solution whose ratio to the optimum is in /spl Omega/(1/ logn). This quadratic programming problem can be seen as an extension to that of maximizing x/sup T/Ay (where y's components are also /spl plusmn/1). Grothendieck's inequality states that the ratio of the optimum value of the latter problem to the optimum of its canonical semidefinite relaxation is bounded below by a constant. The study of this type of quadratic program arose from a desire to approximate the maximum correlation in correlation clustering. Nothing substantive was known about this problem; we present an /spl Omega/ (1/logn) approximation, based on our quadratic programming algorithm. We can also guarantee that our quadratic programming algorithm returns a solution to the MAXCUT problem that has a significant advantage over a random assignment.
The Grothendieck constant KG is the smallest constant such that for every d ∈ N and every matrix A = (aij), sup u i,v j ∈B (d) X aij〈ui, vj 〉 � KG · ij sup x i,y j ∈[−1,1] X ij aijxiyj, where B (d) is the unit ball in R d. Despite several efforts [15, 23], the value of the constant KG remains unknown. The Grothendieck constant KG is precisely the integrality gap of a natural SDP relaxation for the KM,N-Quadratic Programming problem. The input to this problem is a matrix A = (aij) and the objective is to maximize the quadratic form P ij aijxiyj over xi, yj ∈ [−1, 1]. In this work, we apply techniques from [22] to the KM,N-Quadratic Programming problem. Using some standard but non-trivial modifications, the reduction in [22] yields the following hardness result: Assuming the Unique Games Conjecture [9], it is NP-hard to approximate the KM,N-Quadratic Programming problem to any factor better than the Grothendieck constant KG. By adapting a “bootstrapping ” argument used in a proof of Grothendieck inequality [5], we are able to perform a tighter analysis than [22]. Through this careful analysis, we obtain the following new results: ◦ An approximation algorithm for KM,N-Quadratic Programming that is guaranteed to achieve an approximation ratio arbitrarily close to the Grothendieck constant KG (optimal approximation ratio assuming the Unique Games Conjecture). ◦ We show that the Grothendieck constant KG can be computed within an error η, in time depending only on η. Specifically, for each η, we formulate an explicit finite linear program, whose optimum is η-close to the Grothendieck constant. We also exhibit a simple family of operators on the Gaussian Hilbert space that is guaranteed to contain tight examples for the Grothendieck inequality.
The Grothendieck constant KG is the smallest constant such that for every d ∊ ℕ and every matrix A = (aij), where B(d) is the unit ball in ℝd. Despite several efforts [15, 23], the value of the constant KG remains unknown. The Grothendieck constant KG is precisely the integrality gap of a natural SDP relaxation for the KM,N-Quadratic Programming problem. The input to this problem is a matrix A = (αij) and the objective is to maximize the quadratic form Σij aijxiyj over xi, yj ∊ [–1, 1]. In this work, we apply techniques from [22] to the KM,N-Quadratic Programming problem. Using some standard but non-trivial modifications, the reduction in [22] yields the following hardness result: Assuming the Unique Games Conjecture [9], it is NP-hard to approximate the KM,N-Quadratic Programming problem to any factor better than the Grothendieck constant KG. By adapting a “bootstrapping” argument used in a proof of Grothendieck inequality [5], we are able to perform a tighter analysis than [22]. Through this careful analysis, we obtain the following new results: An approximation algorithm for KM,N-Quadratic Programming that is guaranteed to achieve an approximation ratio arbitrarily close to the Grothendieck constant KG (optimal approximation ratio assuming the Unique Games Conjecture). We show that the Grothendieck constant KG can be computed within an error η, in time depending only on η. Specifically, for each η, we formulate an explicit finite linear program, whose optimum is η-close to the Grothendieck constant. We also exhibit a simple family of operators on the Gaussian Hilbert space that is guaranteed to contain tight examples for the Grothendieck inequality.
Following the seminal work of Goemans and Williamson [1], there has been an outgrowth in the use of semidefinite programming (SDP) for designing approximation algorithms. Recall that anα-approximation algorithm for a problem P is a polynomial-time algorithm such that for every instance I of P , it delivers a solution that is within a factor of α of the optimum value [2]. It is well known that SDPs can be solved in polynomial time (up to any prescribed accuracy) via interior-point algorithms (see, e.g., Refs. [3,4]), and they have been used very successfully in the design of approximation algorithms for a host of NP-hard problems, e.g., graph partitioning, graph coloring, and quadratic optimization [1-9], just to name a few.
Grothendieck inequalities are fundamental inequalities which are frequently used in many areas of mathematics and computer science. They can be interpreted as upper bounds for the integrality gap between two optimization problems: a difficult semidefinite program with rank-1 constraint and its easy semidefinite relaxation where the rank constraint is dropped. For instance, the integrality gap of the Goemans-Williamson approximation algorithm for MAX CUT can be seen as a Grothendieck inequality. In this paper we consider Grothendieck inequalities for ranks greater than 1 and we give two applications: approximating ground states in the n-vector model in statistical mechanics and XOR games in quantum information theory.
We prove the following about the Nearest Lattice Vector Problem (in any l/sub p/ norm), the Nearest Code-word Problem for binary codes, the problem of learning a halfspace in the presence of errors, and some other problems. 1. Approximating the optimum within any constant factor is NP-hard. 2. If for some /spl epsiv/>0 there exists a polynomial time algorithm that approximates the optimum within a factor of 2/sup log(0.5-/spl epsiv/)/ /sup n/ then NP is in quasi-polynomial deterministic time: NP/spl sube/DTIME(n/sup poly(log/ /sup n)/). Moreover, we show that result 2 also holds for the Shortest Lattice Vector Problem in the l/sub /spl infin// norm. Improving the factor 2/sup log(0.5-/spl epsiv/)/ /sup n/ to /spl radic/(dim) for either of the lattice problems would imply the hardness of the Shortest Vector Problem in l/sub 2/ norm; an old open problem. Our proofs use reductions from few-prover, one-round interactive proof systems, either directly, or through a set-cover problem.>