Optimal FPT-Approximability for Modular Linear Equations
Abstract
We show optimal FPT-approximability results for solving almost satisfiable systems of modular linear equations, completing the picture of the parameterized complexity and FPT-approximability landscape for the Min--Lin problem for every and . In Min--Lin, we are given a system of linear equations modulo , each on at most variables, and the goal is to find a subset of minimum cardinality such that is satisfiable. The problem is UGC-hard to approximate within any constant factor for every and , which motivates studying it through the lens of parameterized complexity with solution size as the parameter. From previous work (Dabrowski et al. SODA'23/TALG and ESA'25) we know that Min--Lin is W[1]-hard to FPT-approximate within any constant factor when , and that Min--Lin is in FPT when is prime and W[1]-hard when has at least two distinct prime factors. The case when for some prime and has remained an open problem. We resolve this problem in this paper and prove the following: (1) We prove that Min--Lin is in FPT for every prime and . This implies that Min--Lin can be FPT-approximated within a factor of , where is the number of distinct prime factors of . (2) We show that, under the ETH, Min--Lin cannot be FPT-approximated within for any . Our main algorithmic contribution is a new technique coined balanced subgraph covering, which generalizes important balanced subgraphs of Dabrowski et al. (SODA'23/TALG) and shadow removal of Marx and Razgon (STOC'11/SICOMP). For the lower bounds, we develop a framework for proving optimality of FPT-approximation factors under the ETH.
Keywords
Cite
@article{arxiv.2604.10369,
title = {Optimal FPT-Approximability for Modular Linear Equations},
author = {Konrad K. Dabrowski and Peter Jonsson and Sebastian Ordyniak and George Osipov and Magnus Wahlström},
journal= {arXiv preprint arXiv:2604.10369},
year = {2026}
}