The Set of Solutions of Random XORSAT Formulae

Abstract

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). 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 achieved through a low complexity iterative algorithm.

The freezing threshold for k-colourings of a random graph

“…The clusters of k-XOR-SAT are very well-understood, independently by [6,28] (see also [21]). We know the clustering threshold, which in this case is equal to the freezing threshold, and have a very good description of the clusters and the frozen variables.…”

“…Second, much work has gone towards rigorously proving aspects of these hypotheses, eg. [2,3,48,6,28,20,58]. The main contribution of this paper is of the latter type.…”

How does the core sit inside the mantle?

“…Further, in the physics literature it was suggested that the core could be characterised by means of a “message passing” algorithm called Warning Propagation , Chapter 18]. A simple inductive argument shows that the k ‐core coincides with the fixed point of Warning Propagation on In this description of the 2‐core of hypergraphs was used to characterise the clustering phenomenon in the solution space of random XORSAT instances. The Warning Propagation description of the core will play a key role in the present paper.…”

“…In order to study the set of solutions of random k ‐XORSAT formulas Ibrahimi, Kanoria, Kraning and Montanari used Warning Propagation describe the local structure of the 2‐core in random hypergraphs by means of a random marked tree. They showed that the local structure can be described as the limit of Warning Propagation on a suitable Galton‐Watson tree.…”

“…They showed that the local structure can be described as the limit of Warning Propagation on a suitable Galton‐Watson tree. More precisely, , Theorem 3] the analogue of our Lemma 4.4 in the hypergraph setting. The present paper goes one step further by showing that the resulting distribution coincides with the projection of a multi‐type branching process in Section 4.4.…”