19 research outputs found

    Separating the NP-Hardness of the Grothendieck Problem from the Little-Grothendieck Problem

    Get PDF

    Tight hardness of the non-commutative Grothendieck problem

    Get PDF
    We prove that for any ε>0\varepsilon > 0 it is NP-hard to approximate the non-commutative Grothendieck problem to within a factor 1/2+ε1/2 + \varepsilon, which matches the approximation ratio of the algorithm of Naor, Regev, and Vidick (STOC'13). Our proof uses an embedding of ℓ2\ell_2 into the space of matrices endowed with the trace norm with the property that the image of standard basis vectors is longer than that of unit vectors with no large coordinates

    Tight hardness of the non-commutative Grothendieck problem

    Get PDF
    We prove that for any ε > 0 it is NP-hard to approximate the non-commutative Grothendieck problem to within a factor 1=2+ε, which matches the approximation ratio of the algorithm of Naor, Regev, and Vidick (STOC’13). Our proof uses an embedding of ℓ2 into the space of matrices endowed with the trace norm with the property that the image of standard basis vectors is longer than that of unit vectors with no large coordinates. We also observe that one can obtain a tight NP-hardness result for the commutative Little Grothendieck problem; previously, this was only known based on the Unique Games Conjecture (Khot and Naor, Mathematika 2009)

    Grothendieck inequalities for semidefinite programs with rank constraint

    Get PDF
    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 constrained 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.Comment: 22 page

    A Greedy Algorithm for Subspace Approximation Problem

    Get PDF
    In the subspace approximation problem, given m points in R^{n} and an integer k <= n, the goal is to find a k-dimension subspace of R^{n} that minimizes the l_{p}-norm of the Euclidean distances to the given points. This problem generalizes several subspace approximation problems and has applications from statistics, machine learning, signal processing to biology. Deshpande et al. [Deshpande et al., 2011] gave a randomized O(sqrt{p})-approximation and this bound is proved to be tight assuming NP != P by Guruswami et al. [Guruswami et al., 2016]. It is an intriguing question of determining the performance guarantee of deterministic algorithms for the problem. In this paper, we present a simple deterministic O(sqrt{p})-approximation algorithm with also a simple analysis. That definitely settles the status of the problem in term of approximation up to a constant factor. Besides, the simplicity of the algorithm makes it practically appealing
    corecore