Finding densest k-connected subgraphs

Dense subgraph discovery is an important graph-mining primitive with a variety of real-world applications. One of the most well-studied optimization problems for dense subgraph discovery is the densest subgraph problem, where given an edge-weighted undirected graph G = (V, E, w) we are asked to find S ⊆ V that maximizes the density d (S), i.e., half the weighted average degree of the induced subgraph G [S]. This problem can be solved exactly in polynomial time and well-approximately in almost linear time. However, a densest subgraph has a structural drawback, namely, the subgraph may not be robust to vertex/edge failure. Indeed, a densest subgraph may not be well-connected, which implies that the subgraph may be disconnected by removing only a few vertices/edges within it. In this paper, we provide an algorithmic framework to find a dense subgraph that is well-connected in terms of vertex/edge connectivity. Specifically, we introduce the following problems: given a graph G = (V, E, w) and a positive integer/real k, we are asked to find S ⊆ V that maximizes the density d (S) under the constraint that G [S] is k-vertex/edge-connected. For both problems, we propose polynomial-time (bicriteria and ordinary) approximation algorithms, using classic Mader’s theorem in graph theory and its extensions.Accepted manuscrip