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DOI: 10.4230/LIPIcs.STACS.2026.47

On the Hardness of the One-Sided Code Sparsifier Problem

Authors: Elena Grigorescu and Alice Moayyedi

Published in: LIPIcs, Volume 364, 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)

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.

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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)

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@InProceedings{grigorescu_et_al:LIPIcs.STACS.2026.47,
  author =	{Grigorescu, Elena and Moayyedi, Alice},
  title =	{{On the Hardness of the One-Sided Code Sparsifier Problem}},
  booktitle =	{43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
  pages =	{47:1--47:10},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-412-3},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{364},
  editor =	{Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.47},
  URN =		{urn:nbn:de:0030-drops-255365},
  doi =		{10.4230/LIPIcs.STACS.2026.47},
  annote =	{Keywords: Code sparsifiers, NP-hardness, Approximation hardness}
}

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On the Hardness of the One-Sided Code Sparsifier Problem

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