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.

翻译：CSP稀疏化问题探讨：对于给定的CSP实例，在保证所有可能赋值下被满足约束数量保持$1 \pm ε$倍精度的前提下，能否找到最稀疏的权重重置方案？我们开创性地研究随机CSP的稀疏化问题，重点考察两种经典随机模型：$r$-部模型与均匀模型。在$r$-部模型中，变量被划分为$r$个部分，每个约束通过从各部分随机选取一个变量构成；而在均匀模型中，每个约束通过从变量池中随机选取$r$个互异变量形成。对于$r$-部模型，我们揭示了严格的阈值现象：对任意谓词$P$，存在整数$k$使得当$m \le n^k$时$n$个顶点$m$条约束的随机实例本质上无法稀疏化，而当$m \ge n^k$时可稀疏化为约$n^k$规模——此处$k$对应谓词$P$中包含的最大AND结构副本数。这些稀疏化构造具有简洁性，可通过约束的独立同分布采样实现。在均匀模型中，情况更为复杂：对任意谓词$P$，存在整数$k$使得当$m \le n^k$时随机实例无法稀疏化，当$m \ge n^{k+1}$时可稀疏化为约$n^k$规模；但对于某些谓词$P$，当$m \in [n^k, n^{k+1}]$时可能存在或不存在非平凡稀疏化。我们进一步证明存在谓词会使随机实例的稀疏化呈现非单调性，即随着随机约束的增加，实例反而变得更容易稀疏化。最后，我们提出了精确（可高效计算）的判定流程，用于确定特定谓词$P$所属的稀疏化类别。