Limits of Sequential Local Algorithms on the Random $k$-XORSAT Problem

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.