The random $k$-XORSAT problem is a random constraint satisfaction problem of $n$ Boolean variables and $m=rn$ clauses, which a random instance can be expressed as a $G\mathbb{F}(2)$ linear system of the form $Ax=b$, where $A$ is a random $m \times n$ matrix with $k$ ones per row, and $b$ is a random vector. It is known that there exist two distinct thresholds $r_{core}(k) < r_{sat}(k)$ such that as $n \rightarrow \infty$ for $r < r_{sat}(k)$ the random instance has solutions with high probability, while for $r_{core} < r < r_{sat}(k)$ the solution space shatters into an exponential number of clusters. Sequential local algorithms are a natural class of algorithms which assign values to variables one by one iteratively. In each iteration, the algorithm runs some heuristics, called local rules, to decide the value assigned, based on the local neighborhood of the selected variables under the factor graph representation of the instance. We prove that for any $r > r_{core}(k)$ the sequential local algorithms with certain local rules fail to solve the random $k$-XORSAT with high probability. They include (1) the algorithm using the Unit Clause Propagation as local rule for $k \ge 9$, and (2) the algorithms using any local rule that can calculate the exact marginal probabilities of variables in instances with factor graphs that are trees, for $k\ge 13$. The well-known Belief Propagation and Survey Propagation are included in (2). Meanwhile, the best known linear-time algorithm succeeds with high probability for $r < r_{core}(k)$. Our results support the intuition that $r_{core}(k)$ is the sharp threshold for the existence of a linear-time algorithm for random $k$-XORSAT.

翻译：随机$k$-XORSAT问题是一个随机约束满足问题，包含$n$个布尔变量和$m=rn$个子句。其随机实例可表示为$G\mathbb{F}(2)$上的线性方程组$Ax=b$，其中$A$是每行有$k$个1的随机$m \times n$矩阵，$b$是随机向量。已知存在两个不同阈值$r_{core}(k) < r_{sat}(k)$，使得当$n \rightarrow \infty$时，若$r < r_{sat}(k)$则随机实例高概率有解，而当$r_{core} < r < r_{sat}(k)$时解空间破碎成指数多个簇。序列局部算法是一类自然的算法，通过迭代方式逐一为变量赋值。每次迭代中，算法基于实例因子图表示下所选变量的局部邻域，运行称为局部规则的启发式方法以决定赋值的值。我们证明：对于任意$r > r_{core}(k)$，采用特定局部规则的序列局部算法高概率无法求解随机$k$-XORSAT问题。这些算法包括：(1) 当$k \ge 9$时，使用单元子句传播作为局部规则的算法；(2) 当$k \ge 13$时，使用能计算树形因子图实例中变量精确边际概率的任意局部规则的算法。著名的信念传播和调查传播算法属于第(2)类。同时，已知最优线性时间算法在$r < r_{core}(k)$时高概率成功。我们的结果支持$r_{core}(k)$是随机$k$-XORSAT问题存在线性时间算法的尖锐阈值这一直观认识。