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.

The Disjoint Connecting Paths problem and its capacitated generalization, called Unsplittable Flow problem, play an important role in practical applications such as communication network design and routing. These tasks are NP-hard in general, but various polynomial-time approximations are known. Nevertheless, the approximations tend to be either too loose (allowing large deviation from the optimum), or too complicated, often rendering them impractical in large, complex networks. Therefore, our goal is to present a solution that provides a relatively simple, efficient algorithm for the unsplittable flow problem in large directed graphs, where the task is NP-hard, and is known to remain NP-hard even to approximate up to a large factor. The efficiency of our algorithm is achieved by sacrificing a small part of the solution space. This also represents a novel paradigm for approximation. Rather than giving up the search for an exact solution, we restrict the solution space to a subset that is the most important for applications, and excludes only a small part that is marginal in some well-defined sense. Specifically, the sacrificed part only contains scenarios where some edges are very close to saturation. Since nearly saturated links are undesirable in practical applications, therefore, excluding near saturation is quite reasonable from the practical point of view. We refer the solutions that contain no nearly saturated edges as safe solutions, and call the approach safe approximation. We prove that this safe approximation can be carried out efficiently. That is, once we restrict ourselves to safe solutions, we can find the exact optimum by a randomized polynomial time algorithm.

Random graph models play an important role in describing networks with random structural features. The most classical model with the largest number of existing results is the Erd˝os-R´enyi random graph, in which the edges are chosen interdependently at random, with the same probability. In many real-life situations, however, the independence assumption is not realistic. We consider random graphs in which the edges are allowed to be dependent. In our model the edge dependence is quite general, we call it p-robust random graph. Our main result is that for any monotone graph property, the p-robust random graph has at least as high probability to have the property as an Erd˝os-R´enyi random graph with edge probability p. This is very useful, as it allows the adaptation of many results from Erd˝os-R´enyi random graphs to a non-independent setting, as lower bounds.Random networks occur in many practical scenarios. Some examples are wireless ad-hoc networks, various social networks, the web graph describing the World Wide Web, and a multitude of others. Random graph models are often used to describe and analyze such networks. The oldest and most researched random graph model is the Erd˝os-R´enyi random graph G n, p . This denotes a random graph on n nodes, such that each edge is added with probability p, and it is done independently for each edge. A large number of deep results are available about such random graphs, see expositions in the books [1][2][3][4]. Below we list some examples. They are asymptotic results, and for simplicity we ignore rounding issues (i.e., an asymptotic formula may provide a non-integer value for a parameter which is defined as integer for finite graphs). Definition 1 (p-robust random graph)A random graph on n vertices is called p-robust, if every edge is present with probability at least p, regardless of the status (present or not) of other edges. Such a random graph is denoted byNote that p-robustness does not imply independence. It allows that the existence probability of an edge may depend on other edges, possibly in a complicated way, it only requires that the