A General Tractable Density Concept for Graphs

Abstract.

In many applications it is an important algorithmic task to find a densest subgraph in an input graph. The complexity of this task depends on how density is defined. If density means the ratio of the number of edges and the number of vertices in the subgraph, then the algorithmic problem has long been known efficiently solvable. On the other hand, the task becomes NP-hard with closely related but somewhat modified concepts of density. To capture many possible tractable density concepts of interest in a common model, we define and analyze a general concept of density, called F-density. Here F is a family of graphs and we are looking for a subgraph of the input graph, such that this subgraph is the densest in terms of containing the highest number of graphs from F relative to the size of the subgraph. We show that for any fixed finite family F, a subgraph of maximum F-density can be found in polynomial time. As our main tool we develop an algorithm, that may be of independent interest, which can find an independent set of maximum independence ratio in a certain class of weighted graphs. The independence ratio is the weight of the independent set divided by the weight of its neighborhood.