A Characterization of Hard-to-Cover CSPs

We continue the study of covering complexity of constraint satisfaction problems (CSPs) initiated by Guruswami, Hastad and Sudan [SIAM J. Computing, 31(6):1663--1686, 2002] and Dinur and Kol [In Proc. $28$th IEEE Conference on Computational Complexity, 2013]. The covering number of a CSP instance $\Phi$, denoted by $\nu(\Phi)$ 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 regarding how well efficient algorithms can approximate the covering number of a given CSP instance. - Assuming a covering unique games conjecture, introduced by Dinur and Kol, we show that for every non-odd predicate $P$ over any constant sized alphabet and every integer $K$, it is NP-hard to distinguish between $P$-CSP instances (i.e., CSP instances where all the constraints are of type $P$) which are coverable by a constant number of assignments and those whose covering number is at least $K$. Previously, Dinur and Kol, using the same covering unique games conjecture, had shown a similar hardness result for every non-odd predicate over the Boolean alphabet that supports a pairwise independent distribution. Our generalization yields a complete characterization of CSPs over constant sized alphabet $\Sigma$ that are hard to cover since CSP's over odd predicates are trivially coverable with $|\Sigma|$ assignments. - 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 which have covering number at most two and covering number at least $\Omega(\log\log n)$. This generalizes the $4$-LIN result of Dinur and Kol that states it is quasi-NP-hard to distinguish between $4$-LIN-CSP instances which have covering number at most two and covering number at least $\Omega(\log \log\log n)$.