Optimal Inapproximability of Promise Equations over Finite Groups

Silvia Butti; Alberto Larrauri; Stanislav Živný

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arXiv · 2411.01630

Optimal Inapproximability of Promise Equations over Finite Groups

Silvia Butti·Alberto Larrauri·Stanislav Živný

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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.

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Silvia Butti, Alberto Larrauri, Stanislav Živný. 2024-11-03. Optimal Inapproximability of Promise Equations over Finite Groups. https://arxiv.org/abs/2411.01630

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