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Volume 16 (2020) Article 16 pp. 1-30
A Characterization of Hard-to-Cover CSPs
Received: October 13, 2016
Revised: September 10, 2018
Published: December 13, 2020
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Keywords: hardness of approximation, covering complexity, odd predicates
ACM Classification: F.2.2, F.1.3, G.1.6
AMS Classification: 68Q25

Abstract: [Plain Text Version]

$ \newcommand{\twoklin}{\mathsf{2k\text{-}LIN}} \newcommand{\fourlin}{\mathsf{4\text{-}LIN}} $

We continue the study of the covering complexity of constraint satisfaction problems (CSPs) initiated by Guruswami, Håstad and Sudan [SIAM J. Comp. 2002] and Dinur and Kol [CCC'13]. The covering number of a CSP instance $\Phi$ is the smallest number of assignments to the variables of $\Phi$, such that each constraint of $\Phi$ is satisfied by at least one of the assignments. We show the following results:

  1. Assuming a covering variant of the Unique Games Conjecture, introduced by Dinur and Kol, we show that for every non-odd predicate $P$ over any constant-size alphabet and every integer $K$, it is $\mathrm{NP}$-hard to approximate the covering number within a factor of $K$. This yields a complete characterization of CSPs over constant-size alphabets that are hard to cover.

  2. For a large class of predicates that are contained in the $\twoklin$ predicate, we show that it is quasi-$\mathrm{NP}$-hard to distinguish between instances with covering number at most $2$ and those with covering number at least $\Omega(\log\log n)$. This generalizes and improves the $\fourlin$ covering hardness result of Dinur and Kol.

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A preliminary version of this paper appeared in the Proceedings of the 30th Computational Complexity Conference (CCC), 2015.