Sparsification of Two-Variable Valued Constraint Satisfaction Problems

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 reweighted subinstance (V,Π′ ⊂ Π, w′) that preserves the value of the given instance (under every assignment to the variables) within factor 1 ± . 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(|V |/ 2) constraints iff the number of inputs that satisfy P is anything but one (i.e., |P−1(1)| 6= 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.