Tight Bounds for Sparsifying Random CSPs
Joshua Brakensiek, Venkatesan Guruswami, Aaron Putterman
TL;DR
The paper investigates sparsification thresholds for random CSPs in two models, revealing a tight link between sparsifiability and the presence of restricted AND-structures in the predicate. It introduces a sandwiching framework and a decomposition into irrelevant coordinates to overcome worst-case obstructions, yielding sharp, computable exponents that determine sparsifier size in both the random $r$-partite and random uniform models. For Boolean and arbitrary-domain CSPs, as well as for valued (VCSP) CSPs, the authors establish phase transitions at exponents derived from the largest feasible AND, with monotone behavior in the $r$-partite model and nuanced, sometimes non-monotone, behavior in the uniform model. The results extend beyond worst-case guarantees, providing instance-optimal insights, efficient procedures to classify predicates, and a comprehensive translation from complete to random-instance sparsifiability. Collectively, the work delivers a unified, constructive theory for the sparsification of random CSPs with broad implications for algorithm design and understanding average-case CSP complexity.
Abstract
The problem of CSP sparsification asks: for a given CSP instance, what is the sparsest possible reweighting such that for every possible assignment to the instance, the number of satisfied constraints is preserved up to a factor of $1 \pm ε$? We initiate the study of the sparsification of random CSPs. In particular, we consider two natural random models: the $r$-partite model and the uniform model. In the $r$-partite model, CSPs are formed by partitioning the variables into $r$ parts, with constraints selected by randomly picking one vertex out of each part. In the uniform model, $r$ distinct vertices are chosen at random from the pool of variables to form each constraint. In the $r$-partite model, we exhibit a sharp threshold phenomenon. For every predicate $P$, there is an integer $k$ such that a random instance on $n$ vertices and $m$ edges cannot (essentially) be sparsified if $m \le n^k$ and can be sparsified to size $\approx n^k$ if $m \ge n^k$. Here, $k$ corresponds to the largest copy of the AND which can be found within $P$. Furthermore, these sparsifiers are simple, as they can be constructed by i.i.d. sampling of the edges. In the uniform model, the situation is a bit more complex. For every predicate $P$, there is an integer $k$ such that a random instance on $n$ vertices and $m$ edges cannot (essentially) be sparsified if $m \le n^k$ and can sparsified to size $\approx n^k$ if $m \ge n^{k+1}$. However, for some predicates $P$, if $m \in [n^k, n^{k+1}]$, there may or may not be a nontrivial sparsifier. In fact, we show that there are predicates where the sparsifiability of random instances is non-monotone, i.e., as we add more random constraints, the instances become more sparsifiable. We give a precise (efficiently computable) procedure for determining which situation a specific predicate $P$ falls into.