Complexity of finding dense subgraphs

AbstractThe 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 k2 for the k-clique problem. It is shown that (k,f(k))-DSP remains NP-complete for f(k)=Θ(k1+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)=ek2/v2(1+O(vε−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)=ek2/v2(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 (kf(k))-DSP remains for regular graphs: (k,f(k))-DSP is NP-complete for Θ(vε1)-regular graphs if f(k)=Θ(k1+ε2) for any 0<ε1,ε2