Testing for Dense Subsets in a Graph via the Partition Function

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