A celebrated result of Hastad established that, for any constant $\varepsilon>0$, it is NP-hard to find an assignment satisfying a $(1/|G|+\varepsilon)$-fraction of the constraints of a given 3-LIN instance over an Abelian group $G$ even if one is promised that an assignment satisfying a $(1-\varepsilon)$-fraction of the constraints exists. Engebretsen, Holmerin, and Russell showed the same result for 3-LIN instances over any finite (not necessarily Abelian) group. In other words, for almost-satisfiable instances of 3-LIN the random assignment achieves an optimal approximation guarantee. We prove that the random assignment algorithm is still best possible under a stronger promise that the 3-LIN instance is almost satisfiable over an arbitrarily more restrictive group.

翻译：Håstad的一个著名结果表明，对于任意常数$\varepsilon>0$，即使承诺存在满足$(1-\varepsilon)$比例约束的赋值，在阿贝尔群$G$上为给定的3-LIN实例找到满足$(1/|G|+\varepsilon)$比例约束的赋值也是NP难的。Engebretsen、Holmerin和Russell证明了对于任意有限群（未必是阿贝尔群）上的3-LIN实例，同样结论成立。换言之，对于几乎可满足的3-LIN实例，随机赋值算法达到了最优的近似保证。我们证明，在更强的承诺条件下——即3-LIN实例在任意更严格的群上几乎可满足——随机赋值算法仍然是最优的。