Sparsification of Two-Variable Valued Constraint Satisfaction Problems
Arnold Filtser, Robert Krauthgamer · SIAM Journal on Discrete Mathematics · 2017
A valued constraint satisfaction problem (VCSP) instance $(V,\Pi,w)$ is a set of variables $V$ with a set of constraints $\Pi$ weighted by $w$. Given a VCSP instance, we are interested in a reweighted subinstance $(V,\Pi'\subset \Pi,w')$ that preserves the value of the given instance (under every assignment to the variables) within factor $1\pm\epsilon$. 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, $P=\mathsf{XOR}$) can be sparsified into $O(|V|/\epsilon^2)$ constraints iff the number of inputs that satisfy $P$ is anything but one (i.e., $|P^{-1}(1)| eq 1$). 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.