Sparsification of Two-Variable Valued CSPs

A valued constraint satisfaction problem (VCSP) instance $(V,Π,w)$ is a set of variables $V$ with a set of constraints $Π$ weighted by $w$. Given a VCSP instance, we are interested in a re-weighted sub-instance $(V,Π'\subset Π,w')$ such that preserves the value of the given instance (under every assignment to the variables) within factor $1\pmε$. A well-studied special case is cut sparsification in graphs, which has found various applications. We show that a VCSP instance consisting of a single boolean predicate $P(x,y)$ (e.g., for cut, $P=\mbox{XOR}$) can be sparsified into $O(|V|/ε^2)$ constraints if and only if the number of inputs that satisfy $P$ is anything but one (i.e., $|P^{-1}(1)| \neq 1$). Furthermore, this sparsity bound is tight unless $P$ is a relatively trivial predicate. We conclude that also systems of 2SAT (or 2LIN) constraints can be sparsified.