In this paper we analyze the performance of Warning Propagation, a popular message passing algorithm. We show that for 3CNF formulas drawn from a certain distribution over random satisfiable 3CNF formulas, commonly referred to as the planted-assignment distribution, running Warning Propagation in the standard way (run message passing until convergence, simplify the formula according to the resulting assignment, and satisfy the remaining subformula, if necessary, using a simple "off the shelf" heuristic) results in a satisfying assignment when the clause-variable ratio is a sufficiently large constant.
As for relevant results in random graph theory, the seminal work of Alon and Kahale @cite_28 paved the road towards dealing with large-constant-degree planted distributions. @cite_28 present an algorithm that @math @math -colors planted @math -colorable graphs (the distribution of graphs generated by partitioning the @math vertices into @math equally-sized color classes, and including every edge connecting two different color classes with probability @math ; commonly denoted @math ) with a sufficiently large constant expected degree. Building upon the techniques introduced in @cite_28 , Chen and Frieze @cite_14 present an algorithm that 2-colors large constant degree planted 3-uniform bipartite hypergraphs, and Flaxman @cite_7 presents an algorithm for satisfying large-constant clause-variable ratio planted 3SAT instances.
@cite_28 @cite_14 @cite_7
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For many random constraint satisfaction problems, by now there exist asymptotically tight estimates of the largest constraint density for which solutions exist. At the same time, for many of these problems, all known polynomial-time algorithms stop finding 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 achieve this goal. Given the simplicity of those algorithms, one would expect room for improvement. Yet, to date, no algorithm is known that uses (2 - epsiv)chi 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 find it 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 speaking, we prove that the set of k-colorings looks like a giant ball for k ges 2chi, but like an error-correcting code for k les (2 - epsiv)chi. We also prove that an analogous phase transition occurs both in random k-SAT and in random hypergraph 2-coloring. And that for each of these three problems, the location of the transition corresponds to the point where all known polynomial-time algorithms fail. To prove our results we develop a general technique that allows us to establish rigorously much of the celebrated 1-step replica-symmetry-breaking hypothesis of statistical physics for random CSPs. We study a family of closely-related distributed graph problems, which we call degree splitting, where roughly speaking the objective is to partition (or orient) the edges such that each node's degree is split almost uniformly. Our findings lead to answers for a number of problems, a sampling of which includes: • We present a poly log n round deterministic algorithm for (2Δ−1)·(1+o(1))-edge-coloring, where Δ denotes the maximum degree. Modulo the 1 + o(1) factor, this settles one of the long-standing open problems of the area from the 1990's (see e.g. Panconesi and Srinivasan [PODC'92]). Indeed, a weaker requirement of (2Δ − 1) · poly log Δ-edge-coloring in poly log n rounds was asked for in the 4th open question in the Distributed Graph Coloring book by Barenboim and Elkin. • We show that sinkless orientation---i.e., orienting edges such that each node has at least one out-going edge---on Δ-regular graphs can be solved in O(logΔ log n) rounds randomized and in O(logΔ n) rounds deterministically. These prove the corresponding lower bounds by [STOC'16] and Chang, Kopelowitz, and Pettie [FOCS'16] to be tight. Moreover, these show that sinkless orientation exhibits an exponential separation between its randomized and deterministic complexities, akin to the results of for Δ-coloring Δ-regular trees. • We present a randomized O(log4 n) round algorithm for orienting a-arboricity graphs with maximum out-degree a(1 + e). This can be also turned into a decomposition into a(1 + e) forests when a = Ω(log n) and into a(1 + e) pseduo-forests when a = o(log n). Obtaining an efficient distributed decomposition into less than 2a forests was stated as the 10th open problem in the book by Barenboim and Elkin. Consider an n-vertex graph G = (V, E) of maximum degree Δ, and suppose that each vertex v ∈ V hosts a processor. The processors are allowed to communicate only with their neighbors in G. The communication is synchronous, that is, it proceeds in discrete rounds. In the distributed vertex coloring problem, the objective is to color G with Δ + 1, or slightly more than Δ + 1, colors using as few rounds of communication as possible. (The number of rounds of communication will be henceforth referred to as running time.) Efficient randomized algorithms for this problem are known for more than twenty years [ 1986; Luby 1986]. Specifically, these algorithms produce a (Δ + 1)-coloring within O(log n) time, with high probability. On the other hand, the best known deterministic algorithm that requires polylogarithmic time employs O(Δ2) colors. This algorithm was devised in a seminal FOCS’87 paper by Linial [1987]. Its running time is O(log* n). In the same article, Linial asked whether one can color with significantly less than Δ2 colors in deterministic polylogarithmic time. By now, this question of Linial became one of the most central long-standing open questions in this area. In this article, we answer this question in the affirmative, and devise a deterministic algorithm that employs Δ1+o(1) colors, and runs in polylogarithmic time. Specifically, the running time of our algorithm is O(f(Δ)log Δ log n), for an arbitrarily slow-growing function f(Δ) = ω(1). We can also produce an O(Δ1+η)-coloring in O(log Δ log n)-time, for an arbitrarily small constant η > 0, and an O(Δ)-coloring in O(Δe log n) time, for an arbitrarily small constant e > 0. Our results are, in fact, far more general than this. In particular, for a graph of arboricity a, our algorithm produces an O(a1+η)-coloring, for an arbitrarily small constant η > 0, in time O(log a log n). We study the graph coloring problem over random graphs of finite average connectivity c. Given a number q of available colors, we find that graphs with low connectivity admit almost always a proper coloring whereas graphs with high connectivity are uncolorable. Depending on q, we find with a one-step replica-symmetry breaking approximation the precise value of the critical average connectivity @math Moreover, we show that below @math there exists a clustering phase @math in which ground states spontaneously divide into an exponential number of clusters. Furthermore, we extended our considerations to the case of single instances showing consistent results. This leads us to propose a different algorithm that is able to color in polynomial time random graphs in the hard but colorable region, i.e., when @math