Searching for dense subsets in a graph via the partition function

07/05/2018

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by   Alexander Barvinok, et al.

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For a set S of vertices of a graph G, we define its density 0 ≤σ(S) ≤ 1 as the ratio of the number of edges of G spanned by the vertices of S to |S| 2. We show that, given a graph G with n vertices and an integer m, the partition function ∑_S {γ m σ(S) }, where the sum is taken over all m-subsets S of vertices and 0 < γ <1 is fixed in advance, can be approximated within relative error 0 < ϵ < 1 in quasi-polynomial n^O( m - ϵ) time. We discuss numerical experiments and observe that for the random graph G(n, 1/2) one can afford a much larger γ, provided the ratio n/m is sufficiently large.