$ \newcommand{\eps}{\varepsilon} $ We prove that for any $\eps > 0$ it is NP-hard to approximate the non-commutative Grothendieck problem to within a factor $1/2 + \eps$, which matches the approximation ratio of the algorithm of Naor, Regev, and Vidick (STOC'13). Our proof uses an embedding of $\ell_2$ 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. 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). A conference version of this paper appeared in the Proceedings of the 56th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2015).
We prove that for any \eps>0 it is NP-hard to approximate the non-commutative Grothendieck problem to within a factor 1/2+\eps, 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). A conference version of this paper appeared in the Proceedings of the 56th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2015).
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.
Assuming the Unique Games Conjecture, we show it is NP-hard to approximate MAX-3-CUT within a multiplicative factor of α3+ε for every ε>0, where α3≈.83600811464 is the approximation ratio of Frieze-Jerrum's polynomial-time algorithm from 1995. That is, we prove sharp hardness of approximation for MAX-3-CUT. This result resolves a conjecture of Khot-Kindler-Mossel-O'Donnell from 2004 by proving the three candidate Plurality is Stablest Conjecture for correlations in [−1/2,2/5] and generalizes the Majority is Stablest Theorem of Mossel-O'Donnell-Oleszkiewicz [Annals of Math, 2010].
With a similar strategy we prove: assuming the Unique Games Conjecture, it is NP-hard to approximate the product-state value of Quantum MAX-CUT within a multiplicative factor of αBOV+ε for every ε>0, where αBOV≈0.9563372685 is the approximation ratio of the Briët-de Oliveira Filho-Vallentin algorithm. This sharp hardness result completes the conjectured hardness of Hwang-Neeman-Parekh-Thompson-Wright from 2021 by proving their Sk−1-valued Borell inequality for correlations in [−.5843,.5843] for all k≥3.
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.
SVPp is Deterministically NP-Hard for all p>2, Even to Approximate Within a Factor of 2log1−εn
We prove that SVPp is NP-hard to approximate within a factor of 2log1−εn, for all constants ε>0 and p>2, under standard deterministic Karp reductions. This result is also the first proof that \emph{exact} SVPp is NP-hard in a finite ℓp norm. Hardness for SVPp with p finite was previously only known if NP ⊆ RP, and under that assumption, hardness of approximation was only known for all constant factors. As a corollary to our main theorem, we show that under the Sliding Scale Conjecture, SVPp is NP-hard to approximate within a small polynomial factor, for all constants p>2. Our proof techniques are surprisingly elementary; we reduce from a \emph{regularized} PCP instance directly to the shortest vector problem by using simple gadgets related to Vandermonde matrices and Hadamard matrices.
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.