Parameterized Approximability for Modular Linear Equations

2025-09-05

AI Summary

This paper studies the Min-$r$-Lin$(mathbb{Z}_m)$ problem under parameterized approximation: given a system $S$ of linear equations modulo $m$, each of length at most $r$, find a minimum-cardinality subset $Z subseteq S$ such that $S setminus Z$ is satisfiable. For FPT approximation parameterized by the optimum solution size, we introduce a novel framework integrating number-theoretic structure with graph-cut techniques. We establish the first FPT $2$-approximation for Min-2-Lin$(mathbb{Z}_{p^n})$ over prime-power moduli; extend it to arbitrary $m$, achieving approximation ratio $2omega(m)$, where $omega(m)$ denotes the number of distinct prime factors of $m$; and propose an iterative relaxation strategy based on “shadow removal” to overcome hidden unsatisfiability obstacles arising from the non-field ring structure. Furthermore, we prove constant-factor inapproximability lower bounds, revealing inherent hardness for ternary equations and specific rings.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

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
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

We consider the Min-$r$-Lin$(Z_m)$ problem: given a system $S$ of length-$r$ linear equations modulo $m$, find $Z subseteq S$ of minimum cardinality such that $S-Z$ is satisfiable. The problem is NP-hard and UGC-hard to approximate in polynomial time within any constant factor even when $r = m = 2$. We focus on parameterized approximation with solution size as the parameter. Dabrowski et al. showed that Min-$2$-Lin$(Z_m)$ is in FPT if $m$ is prime (i.e. $Z_m$ is a field), and it is W[1]-hard if $m$ is not a prime power. We show that Min-$2$-Lin$(Z_{p^n})$ is FPT-approximable within a factor of $2$ for every prime $p$ and integer $n geq 2$. This implies that Min-$2$-Lin$(Z_m)$, $m in Z^+$, is FPT-approximable within a factor of $2ω(m)$ where $ω(m)$ counts the number of distinct prime divisors of $m$. The idea behind the algorithm is to solve ever tighter relaxations of the problem, decreasing the set of possible values for the variables at each step. Working over $Z_{p^n}$ and viewing the values in base-$p$, one can roughly think of a relaxation as fixing the number of trailing zeros and the least significant nonzero digits of the values assigned to the variables. To solve the relaxed problem, we construct a certain graph where solutions can be identified with a particular collection of cuts. The relaxation may hide obstructions that will only become visible in the next iteration of the algorithm, which makes it difficult to find optimal solutions. To deal with this, we use a strategy based on shadow removal to compute solutions that (1) cost at most twice as much as the optimum and (2) allow us to reduce the set of values for all variables simultaneously. We complement the algorithmic result with two lower bounds, ruling out constant-factor FPT-approximation for Min-$3$-Lin$(R)$ over any nontrivial ring $R$ and for Min-$2$-Lin$(R)$ over some finite commutative rings $R$.

Problem

• Finding minimum unsatisfiable subsets in linear equations modulo m
• Parameterized approximability for modular linear equation systems
• FPT-approximation for Min-2-Lin over prime power moduli

Innovation

• FPT-approximation algorithm for modular linear equations
• Base-p relaxation with shadow removal strategy
• Graph construction identifying solutions via cuts