English

Optimal FPT-Approximability for Modular Linear Equations

Data Structures and Algorithms 2026-04-14 v1

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-rr-Lin(Zm)(\mathbb{Z}_m) problem for every rr and mm. In Min-rr-Lin(Zm)(\mathbb{Z}_m), we are given a system SS of linear equations modulo mm, each on at most rr variables, and the goal is to find a subset ZSZ \subseteq S of minimum cardinality such that SZS - Z is satisfiable. The problem is UGC-hard to approximate within any constant factor for every r2r \geq 2 and m2m \geq 2, 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-rr-Lin(Zm)(\mathbb{Z}_m) is W[1]-hard to FPT-approximate within any constant factor when r3r \geq 3, and that Min-22-Lin(Zm)(\mathbb{Z}_m) is in FPT when mm is prime and W[1]-hard when mm has at least two distinct prime factors. The case when m=pdm = p^d for some prime pp and d2d \geq 2 has remained an open problem. We resolve this problem in this paper and prove the following: (1) We prove that Min-22-Lin(Zpd)(\mathbb{Z}_{p^d}) is in FPT for every prime pp and d1d \geq 1. This implies that Min-22-Lin(Zm)(\mathbb{Z}_{m}) can be FPT-approximated within a factor of ω(m)\omega(m), where ω\omega is the number of distinct prime factors of mm. (2) We show that, under the ETH, Min-22-Lin(Zm)(\mathbb{Z}_m) cannot be FPT-approximated within ω(m)ϵ\omega(m) - \epsilon for any ϵ>0\epsilon > 0. 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}
}
R2 v1 2026-07-01T12:04:37.209Z