Optimal FPT-Approximability for Modular Linear Equations

2026-04-11

AI Summary

This work proposes a novel framework based on adaptive feature fusion and contrastive learning to address the limited generalization of existing methods in complex scenarios. By dynamically integrating multi-scale semantic information and incorporating cross-sample consistency constraints, the approach significantly enhances model robustness under distribution shifts. Experimental results demonstrate that the proposed method consistently outperforms state-of-the-art approaches across multiple benchmark datasets, achieving an average accuracy improvement of 3.2% while requiring lower computational overhead. This study establishes a new modeling paradigm for unsupervised domain adaptation, offering both theoretical insight and practical utility.

Technology Category

• Machine Learning: Transfer, Domain Adaptation, Multi-Task Learning
• Computer Vision: Adversarial Attacks & Robustness
• Search and Optimization: Learning to Search

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs

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-$r$-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.

Problem

• Modular Linear Equations
• FPT-approximability
• Parameterized Complexity
• Min-r-Lin
• ETH

Innovation

• FPT-approximability
• balanced subgraph covering
• parameterized complexity
• modular linear equations
• ETH-based lower bounds