Finding dense subgraphs

Abstract

The dense subgraph problem (DSG) asks, given a graph G and two integers K ₁ and K ₂, whether there is a subgraph of G which has at most K ₁ vertices and at least K ₂ edges. When K ₂=K ₁(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, K ₁, K₂) such that K ₁≤s/2, K ₂≤ K ^1+ε₁ and K ₂ ≤ 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 K ₁=s/2 and K ₂≤e/4(1+O(1/√s)). The condition for K ₂ is quite tight because the answer to DSG is always yes for K ₁=s/2 and k ₂≤e/4(1−O(1/s)). Furthermore there is a deterministic polynomial-time algorithm that finds a subgraph of this density.