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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… (More)

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 Vi-Vj-edges that are actually present in G. We show that on… (More)

We deal with two intimately related subjects: quasi-randomness and regular partitions. The purpose of the concept of quasi-randomness is to measure how much a given graph " resembles " a random one. Moreover , a regular partition approximates a given graph by a bounded number of quasi-random graphs. Regarding quasi-randomness, we present a new spectral… (More)

The minimum bisection problem is to partition the vertices of a graph into two classes of equal size so as to minimize the number of crossing edges. The problem is NP-hard in the worst case. In this paper we analyze a spectral heuristic for the minimum bisection problem on random graphs <i>G<inf>n</inf></i>(<i>p,p'</i>), which are made up as follows.… (More)

Let Φ be a uniformly distributed random k-SAT formula with n variables and m clauses. We present a polynomial time algorithm that finds a satisfying assignment of Φ with high probability for constraint densities m/n < (1 − ε k)2 k ln(k)/k, where ε k → 0. Previously no efficient algorithm was known to find solutions with non-vanishing probability beyond m/n… (More)

Let A be a 0/1 matrix of size m × n, and let p be the density of A (i.e., the number of ones divided by m · n). We show that A can be approximated in the cut norm within ε · mnp by a sum of cut matrices (of rank 1), where the number of summands is independent of the size m·n of A, provided that A satisfies a certain boundedness condition. This decomposition… (More)

Since the early 2000s physicists have developed an ingenious but non-rigorous formalism called the <i>cavity method</i> to put forward precise conjectures as to the phase transitions in random constraint satisfaction problems ("CSPs"). The cavity method comes in two versions: the simpler <i>replica symmetric</i> variant, and the more intricate <i>1-step… (More)

It is exponentially unlikely that a sparse random graph or hypergraph is connected, but such graphs occur commonly as the giant components of larger random graphs. This simple observation allows us to estimate the number of connected graphs, and more generally the number of connected d-uniform hypergraphs, on n vertices with m = O(n) edges. We also estimate… (More)