On the Hardness of the One-Sided Code Sparsifier Problem
Elena Grigorescu, Alice Moayyedi
TL;DR
This paper studies the computational hardness of finding minimal one-sided $\tfrac{1}{2}$-sparsifiers for binary linear codes, showing that the Minimal One-Sided $\tfrac{1}{2}$-Sparsifier problem is NP-hard and inherits strong inapproximability via reductions from the Nearest Codeword Problem (NCP). It develops a coset-based framework with notions of $\alpha$-thick/$\alpha$-thin sets and a structural connection that ties $\tfrac{1}{2}$-thick coset representatives to nearest codewords, enabling reductions from NCP to the sparsifier problem. A constructive polynomial-time reduction, together with known NCP hardness of approximation, yields hardness results for exact and approximate sparsifiers, including gap-type and multiplicative-approximation variants. These results establish fundamental limits on code sparsification, paralleling classical hardness results for nearest codeword problems and underscoring the computational infeasibility of optimal one-sided sparsifiers in general settings.
Abstract
The notion of code sparsification was introduced by Khanna, Putterman and Sudan (arxiv.2311.00788), as an analogue to the the more established notion of cut sparsification in graphs and hypergraphs. In particular, for $α\in (0,1)$ an (unweighted) one-sided $α$-sparsifier for a linear code $\mathcal{C} \subseteq \mathbb{F}_2^n$ is a subset $S\subseteq [n]$ 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 $n/2+O(\sqrt{kn})$ for any linear code, where $k$ is the dimension of $\mathcal{C}$. 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.