Sparsification of Two-Variable Valued CSPs

A valued constraint satisfaction problem (VCSP) instance $(V,\mathrm{\Pi },w)$ is a set of variables $V$ with a set of constraints $\mathrm{\Pi }$ weighted by $w$. Given a VCSP instance, we are interested in a re-weighted sub-instance $(V,\Pi’\subset \Pi,w’)$ such that preserves the value of the given instance (under every assignment to the variables) within factor $1\pm ϵ$. 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, $P=XOR$) can be sparsified into $O(\mathrm{\mid }V\mathrm{\mid }\mathrm{/}{ϵ}^2)$ constraints if and only if the number of inputs that satisfy $P$ is anything but one (i.e., $\mathrm{\mid }{P}^{-1}(1)\mathrm{\mid }\mathrm{\ne }1$). 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.