Performance of Sequential Local Algorithms for the Random NAE-$K$-SAT Problem

Abstract

We formalize the class of "sequential local algorithms" and show that these algorithms fail to find satisfying assignments on random instances of the "Not-All-Equal-$K$-SAT" (NAE-$K$-SAT) problem if the number of message passing iterations is bounded by a function moderately growing in the number of variables and if the clause-to-variable ratio is above $(1+o_K(1)){2^{K-1}\over K}\ln^2 K$ for sufficiently large $K$. Sequential local algorithms are those that iteratively set variables based on some local information and/or local randomness and then recurse on the reduced instance. Our model captures some weak abstractions of natural algorithms such as Survey Propagation (SP)-guided as well as Belief Propagation (BP)-guided decimation algorithms---two widely studied message-passing--based algorithms---when the number of message-passing rounds in these algorithms is restricted to be growing only moderately with the number of variables. The approach underlying our paper is based on an intricate geometry of the solution space of a random NAE-$K$-SAT problem. We show that above the $(1+o_K(1)){2^{K-1}\over K}\ln^2 K$ threshold, the overlap structure of $m$-tuples of nearly (in an appropriate sense) satisfying assignments exhibit a certain behavior expressed in the form of some constraints on pairwise distances between the $m$ assignments for appropriately chosen positive integer $m$. 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 [D. Gamarnik and M. Sudan, Ann. Probab., to appear], where a similar approach was used in a (somewhat simpler) setting of nonsequential local algorithms, this result is the first work that directly links the overlap property of random constraint satisfaction problems to the computational hardness of finding satisfying assignments.