Sparsification of Two-Variable Valued CSPs

TL;DR

This paper investigates sparsification of valued constraint satisfaction problems (VCSPs), establishing tight bounds for when such problems can be reduced in size while approximately preserving their solutions, with implications for 2SAT and 2LIN systems.

Contribution

It characterizes exactly which VCSPs with a single boolean predicate can be sparsified efficiently, providing tight bounds and extending results to 2SAT and 2LIN.

Findings

VCSPs with certain predicates can be sparsified to O(|V|/ε^2) constraints

Sparsification is possible if and only if the predicate's satisfying inputs are not exactly one

The bounds are tight for non-trivial predicates

Abstract

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 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 $1\pm\epsilon$. 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, $P=\mbox{XOR}$) can be sparsified into $O(|V|/\epsilon^2)$ 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.