The algebraic structure of the densification and the sparsification tasks for CSPs

The tractability of certain CSPs for dense or sparse instances is known from the 90s. Recently, the densification and the sparsification of CSPs were formulated as computational tasks and the systematical study of their computational complexity was initiated. We approach this problem by introducing the densification operator, i.e. the closure operator that, given an instance of a CSP, outputs all constraints that are satisfied by all of its solutions. According to the Galois theory of closure operators, any such operator is related to a certain implicational system (or, a functional dependency) $\Sigma$. We are specifically interested in those classes of fixed-template CSPs, parameterized by constraint languages $\Gamma$, for which the size of an implicational system $\Sigma$ is a polynomial in the number of variables $n$. We show that in the Boolean case, $\Sigma$ is of polynomial size if and only if $\Gamma$ is of bounded width. For such languages, $\Sigma$ can be computed in log-space or in a logarithmic time with a polynomial number of processors. Given an implicational system $\Sigma$, the densification task is equivalent to the computation of the closure of input constraints. The sparsification task is equivalent to the computation of the minimal key. This leads to ${\mathcal O}({\rm poly}(n)\cdot N^2)$-algorithm for the sparsification task where $N$ is the number of non-redundant sparsifications of an original CSP. Finally, we give a complete classification of constraint languages over the Boolean domain for which the densification problem is tractable.