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 ksatisfiability (k-SAT), hypergraph bicoloring and graph coloring. 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 intimately tied to the construction of specific subgraphs of the hypergraph associated with an instance of k-XORSAT. In order to study such subgraphs, we establish novel local weak convergence results for them.

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 [22,Chapter 18]. A simple inductive argument shows that the k-core coincides with the fixed point of Warning Propagation on G. In [14] 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 [14] used Warning Propagation describe the local structure of the 2core 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, [14,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.…”

“…There is a very natural formulation of Warning Propagation to identify the k-core of a given graph G, [14,22]. It is based on introducing "messages" on the edges of G and marks on the vertices of G, both with values in {0, 1}.…”

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“…In fact, in a recent paper [10] we combined Warning Propagation with a fixed point analysis on Galton-Watson trees to the k-core problem. A similar approach was used in [24] in the context of random constraint satisfaction problems. Further, in [4] Warning Propagation was applied to the random graph coloring problem.…”