Finding dense subgraphs
The dense subgraph problem (DSG) asks, given a graph G and two integers K1 and K2, whether there is a subgraph of G which has at most K1 vertices and at least K2 edges. When K2=K1(K1−1)/2, DSG is equivalent to well-known CLIQUE. The main purpose of this paper is to discuss the problem of finding slightly dense subgraphs. It is shown that DSG remains NP-complete for the set of instances (G, K1, K2) such that K1≤s/2, K2≤ K 1 1+ε and K2 ≤ e/4(1+9/20+o(1)), where s is the number of G's vertices and e is the number of G's edges. If the second restriction is removed, then the third restriction can be strengthened, i.e., DSG is NP-complete for K1=s/2 and K2≤e/4(1+O(1/√s)). The condition for K2 is quite tight because the answer to DSG is always yes for K1=s/2 and k2≤e/4(1−O(1/s)). Furthermore there is a deterministic polynomial-time algorithm that finds a subgraph of this density.
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