Finding Connected Subgraphs of Fixed Minimum Density: Implementation and Experiments

We consider the following problem. Given a graph and a rational numberi¾?$$\mu $$, $$0 < \mu \le 1$$, find a connected subgraph of density at leasti¾?$$\mu $$ with the largest number of vertices. Here, the density of an $$n$$-vertex graph with $$m$$i¾?edges is $$m/\left {\begin{array}{c}n\\ 2\end{array}}\right $$. This problem arises in many application contexts such as community detection in social networks. We implement a branch and bound algorithm and tune it for efficiency on sparse real-world graphs for the case $$\mu \ge 1/2$$. Central issues for the implementation are the choice of branching candidates, two new upper bounding procedures, and several data reduction and early termination rules.