Theory of Computing
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Title : A Characterization of Hard-to-Cover CSPs
Authors : Amey Bhangale, Prahladh Harsha, and Girish Varma
Volume : 16
Number : 16
Pages : 1-30
URL : https://theoryofcomputing.org/articles/v016a016
Abstract
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We continue the study of the _covering complexity_ of constraint
satisfaction problems (CSPs) initiated by Guruswami, Hastad 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 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
2k-LIN predicate, we show that it is quasi-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 4-LIN covering hardness
result of Dinur and Kol.