Dense subgraph problem revisited

We note the condition 0 ≤ w(e) ≤ 1 is given since it is convenient for presenting our theoretical results, although we can define each problem without this condition. We say unweighted dense subgraph problem if w(e) = 1 for each edge. We define the density ∆ of the output subgraph H to be ∆ = 2w(F ) k(k−1) for the non-bipartite case and ∆ = w(F ) m′n′ for the bipartite case. In other words, density is the ratio of the weight sum to the number of edges in a clique (or a bipartite clique) of the same size. We mainly consider the case where the density of the optimal output subgraph is high (e.g., ∆ = Ω(1)), and aim to design efficient approximation algorithms.