Optimal inapproximability of satisfiable k-LIN over non-abelian groups

A seminal result of Håstad [Hås01] shows that it is NP-hard to find an assignment that satisfies 1 |G| + ε fraction of the constraints of a given k-LIN instance over an abelian group, even if there is an assignment that satisfies (1ε) fraction of the constraints, for any constant ε > 0. Engebretsen et al. [EHR04] later showed that the same hardness result holds for k-LIN instances over any finite non-abelian group.

Unlike the abelian case, where we can efficiently find a solution if the instance is satisfiable, in the non-abelian case, it is NP-complete to decide if a given system of linear equations is satisfiable or not, as shown by Goldmann and Russell [GR02].

Surprisingly, for certain non-abelian groups G, given a satisfiable k-LIN instance over G, one can in fact do better than just outputting a random assignment using a simple but clever algorithm. The approximation factor achieved by this algorithm varies with the underlying group. In this paper, we show that this algorithm is optimal by proving a tight hardness of approximation of satisfiable k-LIN instance over any non-abelian G, assuming P = NP.

As a corollary, we also get 3-query probabilistically checkable proofs with perfect completeness over large alphabets with improved soundness.