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Strong Inapproximability for a Promise Rank Problem

Authors Venkatesan Guruswami , Xuandi Ren , Shaoxuan Tang

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Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
  • Theory of computation → Approximation algorithms analysis
Keywords
  • rank minimization
  • inapproximability
  • promise problems
  • PCP
  • moment matrices

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Abstract

Given a linear subspace of n × n matrices over 𝔽_{2^r} that is promised to contain a matrix of rank 1, we prove that it is hard to find a matrix of rank n^o(1/log log n), assuming NP doesn't have sub-exponential algorithms. In addition to being a basic problem, the hardness of this problem, even for the exact version, drove recent PCP-free inapproximability results for minimum distance and shortest vector problems concerning codes and lattices.
The proof combines the concept of superposition soundness introduced by Khot and Saket with moment matrices. To produce a rank-gap of 1 vs. k, the reduction runs in time n^O(log k). We also give another moment-matrix-based construction which runs in time n^O(k) but works for any finite field F_q.

Cite As Get BibTex

Venkatesan Guruswami, Xuandi Ren, and Shaoxuan Tang. Strong Inapproximability for a Promise Rank Problem. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 19:1-19:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2026.19

Author Details

Venkatesan Guruswami
  • Simons Institute for the Theory of Computing, University of California, Berkeley, CA, USA
  • Departments of EECS and Mathematics, University of California, Berkeley, CA, USA
Xuandi Ren
  • Department of EECS, University of California, Berkeley, CA, USA
Shaoxuan Tang
  • Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing, China

Funding

  • Guruswami, Venkatesan: Research supported in part by NSF grant CCF-2211972, a DARPA grant under Contract No. HR0011262E031, and a Simons Investigator award.
  • Ren, Xuandi: Supported in part by NSF grant CCF-2228287 and V.G.’s Simons Investigator award.

Acknowledgements

Some of this work was done while Shaoxuan Tang was visiting UC Berkeley. The assistance of ChatGPT was used with some of the proofs in Section 4. All proofs were carefully verified by the authors.

References

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