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.

翻译：约束满足问题由取值于有限域的一组变量及这些变量上的局部约束组成，其目标是寻找一个变量赋值，使得满足约束的比例最大化。本文研究约束为有限群G上广义线性方程的约束满足问题。具体而言，对于给定的$S \subseteq G$，该问题中的约束形如变量值的加法（对于非阿贝尔群则为乘法）结果属于集合$S$。我们针对该问题的可满足实例给出了一种近似算法，并证明在假设$P\neq NP$下，该算法对某些$S$而言是最优的。该自然谓词是极少数已知的、在假设$P\neq NP$下对几乎可满足实例具有近似抵抗性、同时却在可满足实例上存在非平凡近似算法的谓词之一。