1 Ju l 2 00 8 Finding Dense Subgraphs in G ( n , 1 / 2 )
نویسندگان
چکیده
Finding the largest clique is a notoriously hard problem, even on random graphs. It is known that the clique number of a random graph G(n, 1/2) is almost surely either k or k + 1, where k = ⌈2 log n − 2 log log n − 1⌉ (Section 4.5 in [1], also [2]). However, a simple greedy algorithm finds a clique of size only log n (1 + o(1)), with high probability, and finding larger cliques – that of size even (1 + ǫ) log n – in randomized polynomial time has been a long-standing open problem [3]. In this paper, we study the following generalization: given a random graph G(n, 1/2) find the largest subgraph with edge density at least (1 − δ). We show that a simple modification of the greedy algorithm finds a subset of 2 log n vertices whose induced subgraph has edge density at least 0.951, with high probability. To complement this, we show that almost surely there is no subset of 2.784 log n vertices whose induced subgraph has edge density 0.951 or more. We use G(n, p) to denote a random graph on n vertices where each pair of vertices appears as an edge independently with probability p. We use V to denote its set of vertices and E to denote its set of edges. Moreover, given two subsets S ⊆ V and T ⊆ V , we use E(S, T ) to denote the set of edges with one endpoint in S and another endpoint in T . The density of the subgraph induced by vertices in S is given by
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تاریخ انتشار 2008