The Densest Subgraph Problem with a Convex/Concave Size Function

we propose a linear-programming-based polynomial-time exact algorithm. It should be emphasized that this algorithm obtains not only an optimal solution to the problem but also subsets of vertices corresponding to the extreme points of the upper convex hull of {(|S|, w(S)) | S subseteq V }, which we refer to as the dense frontier points. We also propose a flow-based combinatorial exact algorithm for unweighted graphs that runs in O(n^3) time. Finally, we propose a nearly-linear-time 3-approximation algorithm