Optimal Inapproximability of Generalized Linear Equations over a Finite Group

2026-05-11

AI Summary

This study addresses constraint satisfaction problems (CSPs) over finite groups with constraints given by generalized linear equations, focusing on efficiently approximating solutions for fully satisfiable instances and characterizing their optimal inapproximability. By integrating structural analysis of finite group algebras with complexity-theoretic reductions, the work identifies a natural class of predicates that are approximation-resistant under near-satisfiability but admit nontrivial approximation algorithms when instances are completely satisfiable. For specific constraint sets \( S \), the paper presents an algorithm whose approximation ratio is optimal under the assumption that P≠NP, thereby bridging a crucial gap in the landscape of CSP approximation complexity between satisfiable instances and the existence of effective approximation algorithms.

Abstract

Constraint satisfaction problems (CSPs) consist of a set of variables taking values from some finite domain and a set of local constraints on these variables. The objective is to find an assignment to the variables that maximizes the fraction of satisfied constraints. In this work, we study the CSP where the constraints are generalized linear equations over a finite group G. More specifically, for a given $S \subseteq G$, the constraints in this CSP are of the form addition of the values to the variables (similarly, product for non-abelian groups), belonging to the set $S$. We give an approximation algorithm for this problem on satisfiable instances and show that it is optimal for certain $S$ assuming $P\neq NP$. This natural predicate is one of the very few known predicates that are approximation resistant on almost satisfiable instances, assuming $P\neq NP$, but admits a non-trivial approximation algorithm on satisfiable instances.

Problem

• constraint satisfaction problem
• generalized linear equations
• finite group
• approximation algorithm
• approximation resistance

Innovation

• constraint satisfaction problem
• generalized linear equations
• finite group
• approximation algorithm
• approximation resistance