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)=Θ(k ^1+ε) where ε may be any constant such that 0<ε<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 ^ε-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 Θ(vε ₁)- regular graphs if f(k)=Θ(k ^1+ε2) for any 0<ε ₁,ε ₂<1.