Tight Bounds for Sparsifying Random CSPs

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.

💡 Research Summary

The paper “Tight Bounds for Sparsifying Random CSPs” investigates the sparsification problem for constraint satisfaction problems (CSPs) in two natural random models: the r‑partite model and the uniform model. Sparsification asks for the smallest possible re‑weighting of constraints such that, for every assignment of the variables, the total weight of satisfied constraints is preserved within a factor of 1 ± ε. While prior work has characterized worst‑case sparsifiability using the combinatorial notion of non‑redundancy, this work focuses on the typical behavior of random instances, providing exact thresholds and tight upper and lower bounds.

Key Concepts

• Largest AND‑restriction (c): For a given predicate P (or valued relation R), c is the maximum arity of an AND‑type sub‑predicate that can be obtained by fixing some variables to constants 0 or 1 (no variable identification). This parameter governs the sparsifiability of random CSPs.
• r‑partite model: Variables are divided into r disjoint parts; each constraint picks one variable from each part uniformly at random.
• Uniform model: Each constraint is formed by choosing r distinct variables uniformly from the whole variable set.

Main Results – r‑partite Model

1. Lower bound: If the number of constraints m is o(n^c), then with high probability no (1 ± ε)‑sparsifier of size o(min{m, n^c}) exists. The proof uses probabilistic concentration (Chernoff/Azuma) to show that any sub‑instance containing the maximal AND‑restriction forces the sparsifier to retain essentially all its constraints.
2. Upper bound: If m ≥ Ω(n^c), an extremely simple sparsifier—obtained by i.i.d. edge sampling—produces a (1 ± ε)‑approximation of size O(n^c · poly(1/ε)). Thus the threshold for sparsifiability is sharp at Θ(n^c).

The result demonstrates a strict phase transition: below n^c no non‑trivial sparsification is possible; above n^c the instance can be reduced to Θ(n^c) constraints, regardless of the total number of constraints. The presence of an AND‑restriction of size c is the sole determinant.

Main Results – Uniform Model

1. Lower bound: For m ≤ o(n^c) no sparsifier of size o(min{m, n^c}) exists (same reasoning as above).
2. Upper bound: For m ≥ Ω(n^{c+1}) a sparsifier of size O(n^c) can be obtained via i.i.d. sampling.
3. Intermediate regime: When m ∈