Abstract. For many random Constraint Satisfaction Problems, by now, we have asymptotically tight estimates of the largest constraint density for which they have solutions. At the same time, all known polynomial-time algorithms for many of these problems already completely fail to find solutions at much smaller densities. For example, it is well-known that it is easy to color a random graph using twice as many colors as its chromatic number. Indeed, some of the simplest possible coloring algorithms already achieve this goal. Given the simplicity of those algorithms, one would expect there is a lot of room for improvement. Yet, to date, no algorithm is known that uses (2 − ǫ)χ colors, in spite of efforts by numerous researchers over the years. In view of the remarkable resilience of this factor of 2 against every algorithm hurled at it, we believe it is natural to inquire into its origin. We do so by analyzing the evolution of the set of k-colorings of a random graph, viewed as a subset of {1, . . . , k} n , as edges are added. We prove that the factor of 2 corresponds in a precise mathematical sense to a phase transition in the geometry of this set. Roughly, the set of k-colorings looks like a giant ball for k ≥ 2χ, but like an error-correcting code for k ≤ (2 − ǫ)χ. We prove that a completely analogous phase transition also occurs both in random k-SAT and in random hypergraph 2-coloring. And that for each problem, its location corresponds precisely with the point were all known polynomial-time algorithms fail. To prove our results we develop a general technique that allows us to prove rigorously much of the celebrated 1-step Replica-Symmetry-Breaking hypothesis of statistical physics for random CSPs.

For various random constraint satisfaction problems there is a significant gap between the largest constraint density for which solutions exist and the largest density for which any polynomial time algorithm is known to find solutions. Examples of this phenomenon include random k‐SAT, random graph coloring, and a number of other random constraint satisfaction problems. To understand this gap, we study the structure of the solution space of random k‐SAT (i.e., the set of all satisfying assignments viewed as a subgraph of the Hamming cube). We prove that for densities well below the satisfiability threshold, the solution space decomposes into an exponential number of connected components and give quantitative bounds for the diameter, volume, and number.© 2010 Wiley Periodicals, Inc. Random Struct. Alg., 38, 251–268, 2011

Vindicating a sophisticated but non-rigorous physics approach called the cavity method, we establish a formula for the mutual information in statistical inference problems induced by random graphs and we show that the mutual information holds the key to understanding certain important phase transitions in random graph models. We work out several concrete applications of these general results. For instance, we pinpoint the exact condensation phase transition in the Potts antiferromagnet on the random graph, thereby improving prior approximate results [Contucci et al.: Communications in Mathematical Physics 2013]. Further, we prove the conjecture from [Krzakala et al.: PNAS 2007] about the condensation phase transition in the random graph coloring problem for any number q ≥ 3 of colors. Moreover, we prove the conjecture on the information-theoretic threshold in the disassortative stochastic block model [Decelle et al.: Phys. Rev. E 2011]. Additionally, our general result implies the conjectured formula for the mutual information in Low-Density Generator Matrix codes [Montanari: IEEE Transactions on Information Theory 2005].intuition on board but require extraneous assumptions (e.g., that the clause length k or the number of colors be very large). Moreover, many proofs require lengthy detours or case analyses that ought to be expendable. Hence, the obvious question is: can we vindicate the physics calculations wholesale?The main result of this paper is that for a wide class of problems within the purview of the replica symmetric cavity method the answer is 'yes'. More specifically, the cavity method reduces a combinatorial problem on a random graph to an optimization problem on the space of probability distributions on a simplex of bounded dimension. We prove that this reduction is valid under a few easy-to-check conditions. Furthermore, we verify that the stochastic optimization problem admits a combinatorial interpretation as the problem of finding an optimal set of Belief Propagation messages on a Galton-Watson tree. Thus, we effectively reduce a problem on a random graph, a mesmerizing object characterized by expansion properties, to a calculation on a random tree. This result reveals an intriguing connection between statistical inference problems and phase transitions in random graph models, specifically a phase transition that we call the information-theoretic threshold, which in many important models is identical to the so-called "condensation phase transition" predicted by physicists [55]. Moreover, the proofs provide a direct rigorous basis for the physics calculations, and we therefore believe that our techniques will find future applications. To motivate the general results about the connection between statistical inference and phase transitions, which we state in Section 2, we begin with four concrete applications that have each received considerable attention in their own right.1.1. The Potts antiferromagnet. As a first example we consider the antiferromagnetic Potts model on the Erdős-Rényi random ...

Abstract. In this paper we study the use of spectral techniques for graph partitioning. Let G = (V, E) be a graph whose vertex set has a "latent" partition V1, . . . , V k . Moreover, consider a "density matrix" E = (Evw)v,w∈V such that for v ∈ Vi and w ∈ Vj the entry Evw is the fraction of all possible ViVj -edges that are actually present in G. We show that on input (G, k) the partition V1, . . . , V k can (almost) be recovered in polynomial time via spectral methods, provided that the following holds: E approximates the adjacency matrix of G in the operator norm, for vertices v ∈ Vi, w ∈ Vj = Vi the corresponding column vectors Ev, Ew are separated, and G is sufficiently "regular" w.r.t. the matrix E . This result in particular applies to sparse graphs with bounded average degree as n = #V → ∞, and it yields interesting consequences on partitioning random graphs.