Performance of the Survey Propagation-guided decimation algorithm for the random NAE-K-SAT problem

Abstract

We show that the Survey Propagation-guided decimation algorithm fails to find satisfying assignments on random instances of the "Not-All-Equal-$K$-SAT" problem if the number of message passing iterations is bounded by a constant independent of the size of the instance and the clause-to-variable ratio is above $(1+o_K(1)){2^{K-1}\over K}\log^2 K$ for sufficiently large $K$. Our analysis in fact applies to a broad class of algorithms described as "sequential local algorithms". Such algorithms iteratively set variables based on some local information and then recurse on the reduced instance. Survey Propagation-guided as well as Belief Propagation-guided decimation algorithms - two widely studied message passing based algorithms, fall under this category of algorithms provided the number of message passing iterations is bounded by a constant. Another well-known algorithm falling into this category is the Unit Clause algorithm. Our work constitutes the first rigorous analysis of the performance of the SP-guided decimation algorithm. The approach underlying our paper is based on an intricate geometry of the solution space of random NAE-$K$-SAT problem. We show that above the $(1+o_K(1)){2^{K-1}\over K}\log^2 K$ threshold, the overlap structure of $m$-tuples of satisfying assignments exhibit a certain clustering behavior expressed in the form of constraints on distances between the $m$ assignments, for appropriately chosen $m$. We further show that if a sequential local algorithm succeeds in finding a satisfying assignment with probability bounded away from zero, then one can construct an $m$-tuple of solutions violating these constraints, thus leading to a contradiction. Along with (citation), this result is the first work which directly links the clustering property of random constraint satisfaction problems to the computational hardness of finding satisfying assignments.