Optimal Inapproximability of Promise Equations over Finite Groups

TL;DR

This paper proves that for 3-LIN problems over finite groups, the simple random assignment approach is optimally inapproximable, even under stronger promises of near-satisfiability, extending previous hardness results.

Contribution

It establishes the optimality of the random assignment algorithm for 3-LIN over arbitrary finite groups under stronger promise conditions.

Findings

Random assignment is optimal for almost-satisfiable 3-LIN over finite groups.

Hardness results extend to non-Abelian groups.

Strengthens previous inapproximability bounds.

Abstract

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.