On the Hardness of the One-Sided Code Sparsifier Problem

2025-10-03

AI Summary

This paper investigates the minimum-size one-sided 1/2-sparser construction problem for linear codes. The problem is proven to be NP-hard and strongly inapproximable—i.e., no polynomial-time approximation algorithm exists—thereby establishing, for the first time, a fundamental computational hardness lower bound for coding sparsification. The authors achieve this by reducing the classical Nearest Codeword Problem (NCP) to the one-sided sparsification problem, leveraging tools from combinatorial coding theory and computational complexity theory to rigorously prove both NP-hardness and strong inapproximability. This result fills a critical gap in the complexity-theoretic understanding of one-sided sparsification and provides essential theoretical grounding for future work—such as designing efficient heuristic algorithms or identifying tractable variants under restricted settings.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Applications of cryptography
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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 $αin (0,1)$ an (unweighted) one-sided $α$-sparsifier for a linear code $mathcal{C} subseteq mathbb{F}_2^n$ is a subset $Ssubseteq [n]$ 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.

Problem

• Studying computational hardness of minimal code sparsifiers
• Proving NP-hardness via nearest codeword problem reduction
• Establishing hardness of approximation for sparsifier minimization

Innovation

• Proves NP-hardness of minimal sparsifier problem
• Uses reduction from nearest codeword problem
• Shows hardness of approximation for sparsifiers