Local Density

2016

Algorithms

Community detection is an important task in the analysis of biological, social or technical networks. We survey different models of cohesive graphs, commonly referred to as clique relaxations, that are used in the detection of network communities. For each clique relaxation, we give an overview of basic model properties and of the complexity of the problem of finding large cohesive subgraphs under this model. Since this problem is usually NP-hard, we focus on combinatorial fixed-parameter algorithms exploiting typical structural properties of input networks.

Section: Related Workmentioning

confidence: 99%

Multivariate Algorithmics for Finding Cohesive Subnetworks

2016

Algorithms

“…Observe that for µ = 0 every graph is a µ-clique, and that a graph is a 1-clique iff it is a clique. The µ-clique concept was studied for example by Abello et al [1] and is sometimes also referred to as µ-dense graph [13]. We show that-in contrast to s-Defective Clique Editing and Average-sPlex Editing-µ-Clique Editing is W[1]-hard and thus presumably fixedparameter intractable with respect to the parameter k for any fixed 0 < µ < 1.…”

Section: Introductionmentioning

confidence: 91%

Editing Graphs into Disjoint Unions of Dense Clusters

Guo

Kanj

Algorithms and Computation

et al. 2009

Abstract. In the Π-Cluster Editing problem, one is given an undirected graph G, a density measure Π, and an integer k ≥ 0, and needs to decide whether it is possible to transform G by editing (deleting and inserting) at most k edges into a dense cluster graph. Herein, a dense cluster graph is a graph in which every connected component K = (VK , EK) satisfies Π. The well-studied Cluster Editing problem is a special case of this problem with Π :="being a clique". In this work, we consider three other density measures that generalize cliques: 1) having at most s missing edges (s-defective cliques), 2) having average degree at least |VK | − s (average-s-plexes), and 3) having average degree at least µ · (|VK| − 1) (µ-cliques), where s and µ are a fixed integer and a fixed rational number, respectively. We first show that the Π-Cluster Editing problem is NP-complete for all three density measures. Then, we study the fixed-parameter tractability of the three clustering problems, showing that the first two problems are fixed-parameter tractable with respect to the parameter (s, k) and that the third problem is W[1]-hard with respect to the parameter k for 0 < µ < 1.

“…The order of a graph is the number of vertices. For a vertex set S ⊆ V we denote by N (S) := v∈S N (v) \ S the neighborhood of S, and by deg(v) the degree of v. We use G[S] to denote the subgraph induced by S. The degeneracy of a graph G is the smallest integer d such that every induced subgraph of G has at least one vertex with degree at most d. The h-index of a graph G is the maximum integer h such that G contains h vertices of degree at least h. The property of being a µ-clique is not hereditary, but has a "nestedness" property [17]: Every µ-clique G = (V, E) has an induced subgraph G on |V | − 1 vertices that is also a µ-clique. For the relevant notions from parameterized complexity, refer to [7].…”

Section: Preliminariesmentioning

confidence: 99%

“…There are many different definitions of what a dense subgraph is [11,17] and for almost all of these formulations, the corresponding computational problems are NP-hard.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Finding Dense Subgraphs of Sparse Graphs

Parameterized and Exact Computation

Sorge

2012

Abstract. We investigate the computational complexity of the Densestk-Subgraph (DkS) problem, where the input is an undirected graph G = (V, E) and one wants to find a subgraph on exactly k vertices with a maximum number of edges. We extend previous work on DkS by studying its parameterized complexity. On the positive side, we show that, when fixing some constant minimum density µ of the sought subgraph, DkS becomes fixed-parameter tractable with respect to either of the parameters maximum degree and h-index of G. Furthermore, we obtain a fixedparameter algorithm for DkS with respect to the combined parameter "degeneracy of G and |V | − k". On the negative side, we find that DkS is W[1]-hard with respect to the combined parameter "solution size k and degeneracy of G". We furthermore strengthen a previous hardness result for DkS [Cai, Comput. J., 2008] by showing that for every fixed µ, 0 < µ < 1, the problem of deciding whether G contains a subgraph of density at least µ is W[1]-hard with respect to the parameter |V | − k.