On independent sets in random graphs
Coja-Oghlan, Amin and Efthymiou, Charilaos (2011) On independent sets in random graphs. In: Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms, San Francisco, CA, USA, 23 - 25 Jan 2011. Published in: SODA '11 Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms pp. 136-144.Full text not available from this repository.
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.
|Item Type:||Conference Item (Paper)|
|Subjects:||Q Science > QA Mathematics|
|Divisions:||Faculty of Science > Mathematics|
|Journal or Publication Title:||SODA '11 Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms|
|Page Range:||pp. 136-144|
|Access rights to Published version:||Restricted or Subscription Access|
|Conference Paper Type:||Paper|
|Title of Event:||Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms|
|Type of Event:||Other|
|Location of Event:||San Francisco, CA, USA|
|Date(s) of Event:||23 - 25 Jan 2011|
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