Abstract. We consider the problem of recovering a planted partition such as a coloring, a small bisection, or a large cut in an (apart from that) random graph. In the last 30 years many algorithms for this problem have been developed that work provably well on various random graph models resembling the Erdős-Rényi model Gn,m. In these random graph models edges are distributed uniformly, and thus the degree distribution is very regular. By contrast, the recent theory of large networks shows that real-world networks frequently have a significantly different distribution of the edges and hence also a different degree distribution. Therefore, a variety of new types of random graphs have been introduced to capture these specific properties. One of the most popular models is characterized by a prescribed expected degree sequence. We study a natural variant of this model that features a planted partition. Our main result is that there is a polynomial time algorithm for recovering (a large part of) the planted partition in this model even in the sparse case, where the average degree is constant. In contrast to prior work, the input of the algorithm consists only of the graph, i.e., no further parameters of the model (such as the expected degree sequence) are revealed to the algorithm.
We investigate the Laplacian eigenvalues of a random graph G(n, d) with a given expected degree distribution d. The main result is that w.h.p. G(n, d) has a large subgraph core(G(n, d)) such that the spectral gap of the normalized Laplacian of core(G(n, d)) is 1 − c 0d −1/2 min with high probability; here c 0 > 0 is a constant, andd min signifies the minimum expected degree. The result in particular applies to sparse graphs withd min = O(1) as n → ∞. The present paper complements the work of Chung, Lu, and Vu [Internet Mathematics 1, 2003].
A simple first moment argument shows that in a randomly chosen k-SAT formula with m clauses over n boolean variables, the fraction of satisfiable clauses is 1 − 2 −k + o(1) as m/n → ∞ almost surely. In this paper, we deal with the corresponding algorithmic strong refutation problem: given a random k-SAT formula, can we find a certificate that the fraction of satisfiable clauses is 1 − 2 −k + o(1) in polynomial time? We present heuristics based on spectral techniques that in the case k = 3 and m ln(n) 6 n 3/2 , and in the case k = 4 and m Cn 2 , find such certificates almost surely. In addition, we present heuristics for bounding the independence number (resp. the chromatic number) of random k-uniform hypergraphs from above (resp. from below) for k = 3, 4.
We apply techniques from the theory of approximation algorithms to the problem of deciding whether a random k-SAT formula is satisfiable. Let Form n,k,m denote a random k-SAT instance with n variables and m clauses. Using known approximation algorithms for MAX CUT or MIN BISECTION, we show how to certify that Form n,4,m is unsatisfiable efficiently, provided that m Cn 2 for a sufficiently large constant C > 0. In addition, we present an algorithm based on the Lovász ϑ function that decides within polynomial expected time whether Form n,k,m is satisfiable, provided that k is even and m C ·4 k n k/2 . Finally, we present an algorithm that approximates random MAX 2-SAT on input Form n,2,m within a factor of 1 − O(n/m) 1/2 in expected polynomial time, for m Cn.
Abstract. It is a well established fact, that -in the case of classical random graphs like (variants of) Gn,p or random regular graphsspectral methods yield efficient algorithms for clustering (e. g. colouring or bisection) problems. The theory of large networks emerging recently provides convincing evidence that such networks, albeit looking random in some sense, cannot sensibly be described by classical random graphs. A variety of new types of random graphs have been introduced. One of these types is characterized by the fact that we have a fixed expected degree sequence, that is for each vertex its expected degree is given. Recent theoretical work confirms that spectral methods can be successfully applied to clustering problems for such random graphs, tooprovided that the expected degrees are not too small, in fact > log® n. In this case however the degree of each vertex is concentrated about its expectation. We show how to remove this restriction and apply spectral methods when the expected degrees are bounded below just by a suitable constant. Our results rely on the observation that techniques developed for the classical sparse G",p random graph (that is p = c/n) can be transferred to the present situation, when we consider a suitably normalized adjacency matrix: We divide each entry of the adjacency matrix by the product of the expected degrees of the incident vertices. Given the host of spectral techniques developed for Gn,p this observation should be of independent interest.
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