ECCC-Report TR20-130https://eccc.weizmann.ac.il/report/2020/130Comments and Revisions published for TR20-130en-usSun, 06 Sep 2020 07:42:45 +0300
Paper TR20-130
| Optimal Inapproximability of Satisfiable k-LIN over Non-Abelian Groups |
Amey Bhangale,
Subhash Khot
https://eccc.weizmann.ac.il/report/2020/130A seminal result of H\r{a}stad [J. ACM, 48(4):798–859, 2001] shows that it is NP-hard to find an assignment that satisfies $\frac{1}{|G|}+\varepsilon$ fraction of the constraints of a given $k$-LIN instance over an abelian group, even if there is an assignment that satisfies $(1-\varepsilon)$ fraction of the constraints, for any constant $\varepsilon>0$. Engebretsen et al. [Theoretical Computer Science, 312(1):17–45, 2004] 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 Russell and Goldmann [Information and Computation, 178(1):253–262, 2002].
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 {\em optimal} by proving a tight hardness of approximation of satisfiable $k$-LIN instance over {\em any} non-abelian $G$, assuming $P \neq NP$.
As a corollary, we also get $3$-query probabilistically checkable proofs with perfect completeness over large alphabets with improved soundness.Sun, 06 Sep 2020 07:42:45 +0300https://eccc.weizmann.ac.il/report/2020/130