Optimal FPT-Approximability for Modular Linear Equations
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| Format: | Preprint |
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2026
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| _version_ | 1866910122102489088 |
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| author | Dabrowski, Konrad K. Jonsson, Peter Ordyniak, Sebastian Osipov, George Wahlström, Magnus |
| author_facet | Dabrowski, Konrad K. Jonsson, Peter Ordyniak, Sebastian Osipov, George Wahlström, Magnus |
| contents | 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-$r$-Lin$(\mathbb{Z}_m)$ problem for every $r$ and $m$. In Min-$r$-Lin$(\mathbb{Z}_m)$, we are given a system $S$ of linear equations modulo $m$, each on at most $r$ variables, and the goal is to find a subset $Z \subseteq S$ of minimum cardinality such that $S - Z$ is satisfiable. The problem is UGC-hard to approximate within any constant factor for every $r \geq 2$ and $m \geq 2$, 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-$r$-Lin$(\mathbb{Z}_m)$ is W[1]-hard to FPT-approximate within any constant factor when $r \geq 3$, and that Min-$2$-Lin$(\mathbb{Z}_m)$ is in FPT when $m$ is prime and W[1]-hard when $m$ has at least two distinct prime factors. The case when $m = p^d$ for some prime $p$ and $d \geq 2$ has remained an open problem. We resolve this problem in this paper and prove the following:
(1) We prove that Min-$2$-Lin$(\mathbb{Z}_{p^d})$ is in FPT for every prime $p$ and $d \geq 1$. This implies that Min-$2$-Lin$(\mathbb{Z}_{m})$ can be FPT-approximated within a factor of $ω(m)$, where $ω$ is the number of distinct prime factors of $m$.
(2) We show that, under the ETH, Min-$2$-Lin$(\mathbb{Z}_m)$ cannot be FPT-approximated within $ω(m) - ε$ for any $ε> 0$.
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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_10369 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Optimal FPT-Approximability for Modular Linear Equations Dabrowski, Konrad K. Jonsson, Peter Ordyniak, Sebastian Osipov, George Wahlström, Magnus Data Structures and Algorithms 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-$r$-Lin$(\mathbb{Z}_m)$ problem for every $r$ and $m$. In Min-$r$-Lin$(\mathbb{Z}_m)$, we are given a system $S$ of linear equations modulo $m$, each on at most $r$ variables, and the goal is to find a subset $Z \subseteq S$ of minimum cardinality such that $S - Z$ is satisfiable. The problem is UGC-hard to approximate within any constant factor for every $r \geq 2$ and $m \geq 2$, 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-$r$-Lin$(\mathbb{Z}_m)$ is W[1]-hard to FPT-approximate within any constant factor when $r \geq 3$, and that Min-$2$-Lin$(\mathbb{Z}_m)$ is in FPT when $m$ is prime and W[1]-hard when $m$ has at least two distinct prime factors. The case when $m = p^d$ for some prime $p$ and $d \geq 2$ has remained an open problem. We resolve this problem in this paper and prove the following: (1) We prove that Min-$2$-Lin$(\mathbb{Z}_{p^d})$ is in FPT for every prime $p$ and $d \geq 1$. This implies that Min-$2$-Lin$(\mathbb{Z}_{m})$ can be FPT-approximated within a factor of $ω(m)$, where $ω$ is the number of distinct prime factors of $m$. (2) We show that, under the ETH, Min-$2$-Lin$(\mathbb{Z}_m)$ cannot be FPT-approximated within $ω(m) - ε$ for any $ε> 0$. 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. |
| title | Optimal FPT-Approximability for Modular Linear Equations |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2604.10369 |