In Search of the Densest Subgraph

Abstract

In this survey paper, we review various concepts of graph density, as well as associated theorems and algorithms. Our goal is motivated by the fact that, in many applications, it is a key algorithmic task to extract a densest subgraph from an input graph, according to some appropriate definition of graph density. While this problem has been the subject of active research for over half of a century, with many proposed variants and solutions, new results still continuously emerge in the literature. This shows both the importance and the richness of the subject. We also identify some interesting open problems in the field.

Routing for Global Congestion Avoidance

“…There are, however, other measures for which this maximization is NP-hard. See many more examples and related results in [1].…”

“…The efficiency of k-core follows from two facts. (1) It can be found in polynomial time for any k, see [1]. (2) The k-core for any k is unique.…”

“…This density measure characterizes the graph density by the edge connectivity of a subgraph that has maximum edge connectivity. It can be found in polynomial time, see [1]. However, to precisely satisfy Definition 5, we need a polynomial time algorithm that can list all subgraphs that have edge connectivity at least ρ 0 .…”

“…On the other hand, we can still satisfy efficiency in the somewhat weaker sense of Definition 3. The reason is that if we look for maximal size subgraphs with maximum edge-connectivity (they can be found in polynomial time, see [1]), there can only be linearly many of them. The reason is that any two such maximal subgraphs, if they do not coincide, then they are node-disjoint.…”

“…• For a number of density measures it is known that maximizing them is NP-hard, but efficient approximation algorithms with proven approximation guarantees are often available (see [1]). How can our framework be extended to such cases?…”