On the independence number of random graphs
Discrete Mathematics
Analysis of two simple heuristics on a random instance of k-SAT
Journal of Algorithms
Algorithmic theory of random graphs
Random Structures & Algorithms - Special issue: average-case analysis of algorithms
The analysis of a list-coloring algorithm on a random graph
FOCS '97 Proceedings of the 38th Annual Symposium on Foundations of Computer Science
Random Structures & Algorithms
Maximum matching in sparse random graphs
SFCS '81 Proceedings of the 22nd Annual Symposium on Foundations of Computer Science
Algorithmic Barriers from Phase Transitions
FOCS '08 Proceedings of the 2008 49th Annual IEEE Symposium on Foundations of Computer Science
Reconstruction threshold for the hardcore model
APPROX/RANDOM'10 Proceedings of the 13th international conference on Approximation, and 14 the International conference on Randomization, and combinatorial optimization: algorithms and techniques
The Monotone Complexity of k-clique on Random Graphs
FOCS '10 Proceedings of the 2010 IEEE 51st Annual Symposium on Foundations of Computer Science
Independent sets in random graphs from the weighted second moment method
APPROX'11/RANDOM'11 Proceedings of the 14th international workshop and 15th international conference on Approximation, randomization, and combinatorial optimization: algorithms and techniques
Catching the k-NAESAT threshold
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
The freezing threshold for k-colourings of a random graph
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
Limits of local algorithms over sparse random graphs
Proceedings of the 5th conference on Innovations in theoretical computer science
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The independence number of a sparse random graph G(n, m) of average degree d = 2m/n is well-known to be α(G(n, m)) ~ 2n ln(d)/d with high probability. Moreover, a trivial greedy algorithm w.h.p. finds an independent set of size (1 + o(1)) · n ln(d)/d, i.e., half the maximum size. Yet in spite of 30 years of extensive research no efficient algorithm has emerged to produce an independent set with (1 + ε)n ln(d)/d, for any fixed ε 0. In this paper we prove that the combinatorial structure of the independent set problem in random graphs undergoes a phase transition as the size k of the independent sets passes the point k ~ n ln(d)/d. Roughly speaking, we prove that independent sets of size k (1 + ε)n ln(d)/d form an intricately ragged landscape, in which local search algorithms are bound to get stuck. We illustrate this phenomenon by providing an exponential lower bound for the Metropolis process, a Markov chain for sampling independents sets.