Document Open Access Logo

On the Hardness of the One-Sided Code Sparsifier Problem

Authors Elena Grigorescu , Alice Moayyedi

PDF

File

Thumbnail PDF
PDF
LIPIcs.STACS.2026.47.pdf
  • Filesize: 0.75 MB
  • 10 pages

Document Identifiers

Subject Classification

ACM Subject Classification
  • Theory of computation → Error-correcting codes
Keywords
  • Code sparsifiers
  • NP-hardness
  • Approximation hardness

Metrics

Abstract

The notion of code sparsification was introduced by Khanna, Putterman and Sudan (SODA 2024) as an analogue to the more established notion of cut sparsification in graphs and hypergraphs. In particular, for α ∈ (0,1), an (unweighted) one-sided α-sparsifier for a linear code 𝒞 ⊆ 𝐅₂ⁿ is a subset S ⊆ [n] such that the weight of each codeword projected onto the coordinates in S is preserved up to an α fraction. Recently, Gharan and Sahami (arXiv:2502.02799) show the existence of one-sided 1/2-sparsifiers of size n/2+O(√{kn}) for any linear code, where k is the dimension of 𝒞. In this paper, we consider the computational problem of finding a one-sided 1/2-sparsifier of minimal size, and show that it is NP-hard, via a reduction from the classical nearest codeword problem. We also show hardness of approximation results.

Cite As Get BibTex

Elena Grigorescu and Alice Moayyedi. On the Hardness of the One-Sided Code Sparsifier Problem. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 47:1-47:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.STACS.2026.47

Author Details

Elena Grigorescu
  • David R. Cheriton School of Computer Science, University of Waterloo, Canada
Alice Moayyedi
  • David R. Cheriton School of Computer Science, University of Waterloo, Canada

References

  1. Per Austrin and Subhash Khot. A simple deterministic reduction for the gap minimum distance of code problem. IEEE Trans. Inf. Theory, 60(10):6636-6645, 2014. URL: https://doi.org/10.1109/TIT.2014.2340869.
  2. Vijay Bhattiprolu, Venkatesan Guruswami, Euiwoong Lee, and Xuandi Ren. Inapproximability of finding sparse vectors in codes, subspaces, and lattices, 2025. URL: https://arxiv.org/abs/2410.02636.
  3. Vijay Bhattiprolu and Euiwoong Lee. Inapproximability of sparsest vector in a real subspace. Electron. Colloquium Comput. Complex., TR24-151, 2024. URL: https://eccc.weizmann.ac.il/report/2024/151.
  4. Ilya Dumer, Daniele Micciancio, and Madhu Sudan. Hardness of approximating the minimum distance of a linear code. IEEE Trans. Inf. Theory, 49(1):22-37, 2003. URL: https://doi.org/10.1109/TIT.2002.806118.
  5. Shayan Oveis Gharan and Arvin Sahami. Unweighted one-sided code sparsifiers and thin subgraphs, 2025. URL: https://arxiv.org/abs/2502.02799.
  6. David R. Karger. Using randomized sparsification to approximate minimum cuts. In Daniel Dominic Sleator, editor, Proceedings of the Fifth Annual ACM-SIAM Symposium on Discrete Algorithms. 23-25 January 1994, Arlington, Virginia, USA, pages 424-432. ACM/SIAM, 1994. URL: http://dl.acm.org/citation.cfm?id=314464.314582.
  7. Sanjeev Khanna, Aaron (Louie) Putterman, and Madhu Sudan. Code sparsification and its applications. In David P. Woodruff, editor, Proceedings of the 2024 ACM-SIAM Symposium on Discrete Algorithms, SODA 2024, Alexandria, VA, USA, January 7-10, 2024, pages 5145-5168. SIAM, 2024. URL: https://doi.org/10.1137/1.9781611977912.185.
  8. Subhash Khot. Hardness of approximating the shortest vector problem in lattices. J. ACM, 52(5):789-808, 2005. URL: https://doi.org/10.1145/1089023.1089027.
  9. Dmitry Kogan and Robert Krauthgamer. Sketching cuts in graphs and hypergraphs. In Tim Roughgarden, editor, Proceedings of the 2015 Conference on Innovations in Theoretical Computer Science, ITCS 2015, Rehovot, Israel, January 11-13, 2015, pages 367-376. ACM, 2015. URL: https://doi.org/10.1145/2688073.2688093.
  10. Daniele Micciancio. Locally dense codes. In IEEE 29th Conference on Computational Complexity, CCC 2014, Vancouver, BC, Canada, June 11-13, 2014, pages 90-97. IEEE, IEEE Computer Society, 2014. URL: https://doi.org/10.1109/CCC.2014.17.
  11. Alexander Vardy. The intractability of computing the minimum distance of a code. IEEE Trans. Inf. Theory, 43(6):1757-1766, 1997. URL: https://doi.org/10.1109/18.641542.
Any Issues?
X

Feedback on the Current Page

CAPTCHA

Thanks for your feedback!

Feedback submitted to Dagstuhl Publishing

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact