Complexity of finding dense subgraphs

The k-f (k) dense subgraph problem((k, f (k))-DSP) asks whether there is a k-vertex subgraph of a given graph G which has at least f (k) edges. When f (k) = k(k - 1)/2, (k, f (k))-DSP is equivalent to the well-known k-clique problem. The main purpose of this paper is to discuss the problem of finding slightly dense subgraphs. Note that f(k) is about k(2) for the k-clique problem. It is shown that (k, f (k))-DSP remains NP-complete for f (k) = Theta(k(1-epsilon)) where epsilon may be any constant such that 0 < epsilon < 1. It is also NP-complete for "relatively" slightly-dense subgraphs, i.e., (k, f(k))-DSP is NP-complete for f(k) = ek(2)/v(2)(1 +O(v(epsilon-1))), where v is the number of G's vertices and e is the number of G's edges. This condition is quite tight because the answer to (k, f (k))-DSP is always yes for f (k) = ek(2/)v(2) (1 - (v - k)/(vk - k)) that is the average number of edges in a subgraph of k vertices. Also, we show that the hardness of (k, f(k))-DSP remains for regular graphs: (k, f (k))-DSP is NP-complete for Theta(v(epsilon))-regular graphs if f (k) = Theta(k(1+epsilon2)) for any 0 < epsilon(1), epsilon(2) < 1. (C) 2002 Elsevier Science B.V. All rights reserved.