A Characterization of hard-to-cover CSPs

We continue the study of the covering complexity of constraint satisfaction problems (CSPs) initiated by Guruswami, H{\aa}stad and Sudan [SIAM J. Comp. 2002] and Dinur and Kol [CCC’13]. The covering number of a CSP instance $\mathrm{\Phi }$ is the smallest number of assignments to the variables of $\mathrm{\Phi }$, such that each constraint of $\mathrm{\Phi }$ 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 $\mathrm{\Omega }(\log \log n)$. This generalizes and improves the 4-LIN covering hardness result of Dinur and Kol.