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.
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.
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].
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 little Grothendieck problem (a special case of Boolean quadratic optimization) consists of maximizing P ij Cijxixj over binary variables xi 2 f 1g, where C is a positive semidenite matrix. In this paper we focus on a natural generalization of this problem, the little Grothendieck problem over the orthogonal group. Given C 2 R dn dn a positive semidenite matrix, the objective is to maximize P ij tr C T ijOiO T restricting Oi to take values in the group of orthogonal matrices O(d), where Cij denotes the (ij)-th d d block of C. We propose an approximation algorithm, to which we refer as Orthogonal-Cut, to solve the little Grothendieck problem over the group of orthogonal matrices O(d) and show a constant approximation ratio. Our method is based on semidenite programming where the relaxation is inspired by the work of Goemans and Williamson in the context of the MaxCut problem. For a given d 1, we show a constant approximation ratio of 2, where d is the expected average singular value of a d d matrix with random GaussianN 0; 1 i.i.d. entries. For d = 1 we recover the known 2 = 2= approximation guarantee for the classical little Grothendieck problem. Orthogonal-Cut also serves as an approximation algorithm for several applications including the Procrustes problem where it improves over the best previously known approximation ratio of 1 2 p 2 . The little Grothendieck problem falls under the larger class of problems approximated by an algorithm recently proposed in the context of the non-commutative Grothendieck inequality. Nonetheless, our approach is simpler and gives a better approximation ratio.
The quadratic maximization problem with rank constraint means where n and m are two given positive integers with n ≤ m, Q is a given real symmetric matrix, is the unit sphere in and stands for the Euclidean inner product. In this note, motivated by Briët, Filho and Vallentin's and Ye's excellent works [J. Briët, F. Filho, and F. Vallentin, The positive semidefinite Grothendieck problem with rank constraint, Automata, Languages and Programming, Lecture Notes in Computer Science, Vol. 6198, 2010, pp. 31–42.; Y. Ye, Approximating quadratic programming with bound and quadratic constraints, Math. Program. 84 (1999), pp. 219–226], we first derive the relative approximation ratio for the above problem with arbitrary symmetric matrix Q. While Q was supposed to be positive semidefinite in [J. Briët, F. Filhom, and F. Vallentin, The positive semidefinite Grothendieck problem with rank constraint, Automata, Languages and Programming, Lecture Notes in Computer Science, Vol. 6198, 2010, pp. 31–42.]. Secondly, we extend such a relative approximation ratio to a generalization of the above optimization problem. Finally, a slightly improved relative approximation ratio for the optimization problem considered in [Y. Ye, Approximating quadratic programming with bound and quadratic constraints, Math. Program. 84 (1999), pp. 219–226] is presented.
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.
In this paper, we consider a binary quadratic program (BQP) with random objective coefficients. Given only information on the marginal distributions of the objective coefficients, we propose a tight bound on the expected optimal value of the random BQP. We show that the complexity of computing this bound does not increase substantially with respect to the complexity of solving the corresponding deterministic BQP. For the quadratic unconstrained binary optimization (QUBO) problem with nonnegative off-diagonal random entries, the bound is shown to be computable in polynomial time. We generalize the asymptotic bound for the random quadratic assignment problem from independent random variables to dependent random variables and propose a new closed-form bound on the expected optimal value of the quadratic k-cluster problem. We also provide polynomial time computable upper bounds on the expected optimal value for the NP-hard instances using the linear and semidefinite programming relaxation of the deterministic BQP. The semidefinite programming bound of the expected optimal value of the random MAX-CUT problem inherits the approximation ratio of 0.878 from the semidefinite relaxation of the deterministic MAX-CUT problem. Computational experiments on random QUBO problems and random quadratic knapsack problems provide evidence of the quality of the bounds. Overall, our results indicate that the new bound on the expected optimal value of the random BQP is attractive since it exploits many of the nice results and formulations of the deterministic BQP and is valid under limited distributional information.