随机XORSAT公式的解集
The set of solutions of random XORSAT formulae
摘要：
异或可满足性（XORSAT）问题要求找到满足$m$异或（XOR）子句的$n$布尔变量的赋值，由此每个子句约束变量的子集。我们考虑随机XORSAT实例，从包含$n$变量和大小为$k$的$m$子句的公式集合中随机均匀抽取。该模型与其他约束满足问题的集合，如$k$ -可满足性（$k$ -SAT），超图双着色和图着色，具有一些结构上的相似性。对于许多这样的集合，随着每个变量的约束数量的增加，解的集合分解成指数数量的分离良好的分量。这种现象似乎与解决此类问题的随机实例的困难有关。我们证明了随机$k$ -XORSAT的这种聚类相变的完整表征。特别是，我们证明了聚类阈值是尖锐的，并确定其确切的位置。我们证明了一组解决方案具有较大的电导低于此阈值，每个集群具有较大的电导高于相同的阈值。我们的证明构造了一个非常稀疏的基础的解决方案（或集群内的子集）。这种构造与与$k$ -XORSAT实例相关的超图的特定子图的构造密切相关。为了研究这类子图，我们建立了新的局部弱收敛结果。
英文摘要：
The XOR-satisfiability (XORSAT) problem requires finding an assignment of $n$ Boolean variables that satisfy $m$ exclusive OR (XOR) clauses, whereby each clause constrains a subset of the variables. We consider random XORSAT instances, drawn uniformly at random from the ensemble of formulae containing $n$ variables and $m$ clauses of size $k$. This model presents several structural similarities to other ensembles of constraint satisfaction problems, such as $k$-satisfiability ($k$-SAT), hypergraph bicoloring and graph coloring. For many of these ensembles, as the number of constraints per variable grows, the set of solutions shatters into an exponential number of well-separated components. This phenomenon appears to be related to the difficulty of solving random instances of such problems. We prove a complete characterization of this clustering phase transition for random $k$-XORSAT. In particular, we prove that the clustering threshold is sharp and determine its exact location. We prove that the set of solutions has large conductance below this threshold and that each of the clusters has large conductance above the same threshold. Our proof constructs a very sparse basis for the set of solutions (or the subset within a cluster). This construction is intimately tied to the construction of specific subgraphs of the hypergraph associated with an instance of $k$-XORSAT. In order to study such subgraphs, we establish novel local weak convergence results for them.
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