Optimal Inapproximability of Generalized Linear Equations over a Finite Group

TL;DR

This paper investigates the inapproximability of generalized linear equations over finite groups in CSPs, providing optimal algorithms and hardness results under standard complexity assumptions.

Contribution

It introduces an approximation algorithm for satisfiable instances and proves its optimality for certain cases assuming P≠NP.

Findings

Provides an approximation algorithm for the CSP with linear equations over finite groups.

Shows the algorithm's optimality for specific sets S under P≠NP.

Identifies predicates that are approximation resistant yet admit non-trivial 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.