Large independent sets in Markov random graphs
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Computing the maximum size of an independent set in a graph is a famously hard combinatorial problem that has been well-studied for various classes of graphs. When it comes to random graphs, only the classical binomial random graph G_n,p has been analysed and shown to have largest independent sets of size Θ(logn) w.h.p. This classical model does not capture any dependency structure between edges that can appear in real-world networks. We initiate study in this direction by defining random graphs G^r_n,p whose existence of edges is determined by a Markov process that is also governed by a decay parameter r∈(0,1]. We prove that w.h.p. G^r_n,p has independent sets of size (1-r/2+ϵ) n/logn for arbitrary ϵ > 0, which implies an asymptotic lower bound of Ω(π(n)) where π(n) is the prime-counting function. This is derived using bounds on the terms of a harmonic series, Turán bound on stability number, and a concentration analysis for a certain sequence of dependent Bernoulli variables that may also be of independent interest. Since G^r_n,p collapses to G_n,p when there is no decay, it follows that having even the slightest bit of dependency (any r < 1) in the random graph construction leads to the presence of large independent sets and thus our random model has a phase transition at its boundary value of r=1. For the maximal independent set output by a greedy algorithm, we deduce that it has a performance ratio of at most 1 + logn/(1-r) w.h.p. when the lowest degree vertex is picked at each iteration, and also show that under any other permutation of vertices the algorithm outputs a set of size Ω(n^1/1+τ), where τ=1/(1-r), and hence has a performance ratio of O(n^1/2-r).
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