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

Authors Vijay Bhattiprolu, Euiwoong Lee, Madhur Tulsiani

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Vijay Bhattiprolu
  • Institute for Advanced Study, Princeton, NJ, USA
  • Princeton University, NJ, USA
Euiwoong Lee
  • University of Michigan, Ann-Arbor, USA
Madhur Tulsiani
  • Toyota Technological Institute Chicago, IL, USA


We thank the anonymous ITCS'22 referees for suggesting useful corrections to the manuscript.

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Vijay Bhattiprolu, Euiwoong Lee, and Madhur Tulsiani. Separating the NP-Hardness of the Grothendieck Problem from the Little-Grothendieck Problem. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 22:1-22:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


Grothendieck’s inequality [Grothendieck, 1953] states that there is an absolute constant K > 1 such that for any n× n matrix A, ‖A‖_{∞→1} := max_{s,t ∈ {± 1}ⁿ}∑_{i,j} A[i,j]⋅s(i)⋅t(j) ≥ 1/K ⋅ max_{u_i,v_j ∈ S^{n-1}}∑_{i,j} A[i,j]⋅⟨u_i,v_j⟩. In addition to having a tremendous impact on Banach space theory, this inequality has found applications in several unrelated fields like quantum information, regularity partitioning, communication complexity, etc. Let K_G (known as Grothendieck’s constant) denote the smallest constant K above. Grothendieck’s inequality implies that a natural semidefinite programming relaxation obtains a constant factor approximation to ‖A‖_{∞ → 1}. The exact value of K_G is yet unknown with the best lower bound (1.67…) being due to Reeds and the best upper bound (1.78…) being due to Braverman, Makarychev, Makarychev and Naor [Braverman et al., 2013]. In contrast, the little Grothendieck inequality states that under the assumption that A is PSD the constant K above can be improved to π/2 and moreover this is tight. The inapproximability of ‖A‖_{∞ → 1} has been studied in several papers culminating in a tight UGC-based hardness result due to Raghavendra and Steurer (remarkably they achieve this without knowing the value of K_G). Briet, Regev and Saket [Briët et al., 2015] proved tight NP-hardness of approximating the little Grothendieck problem within π/2, based on a framework by Guruswami, Raghavendra, Saket and Wu [Guruswami et al., 2016] for bypassing UGC for geometric problems. This also remained the best known NP-hardness for the general Grothendieck problem due to the nature of the Guruswami et al. framework, which utilized a projection operator onto the degree-1 Fourier coefficients of long code encodings, which naturally yielded a PSD matrix A. We show how to extend the above framework to go beyond the degree-1 Fourier coefficients, using the global structure of optimal solutions to the Grothendieck problem. As a result, we obtain a separation between the NP-hardness results for the two problems, obtaining an inapproximability result for the Grothendieck problem, of a factor π/2 + ε₀ for a fixed constant ε₀ > 0.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational complexity and cryptography
  • Mathematics of computing → Mathematical optimization
  • Mathematics of computing → Functional analysis
  • Grothendieck’s Inequality
  • Hardness of Approximation
  • Semidefinite Programming
  • Optimization


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