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Isolation Concepts for Efficiently Enumerating Dense Subgraphs
, 2009
"... In an undirected graph G = (V,E), a set of k vertices is called cisolated if it has less than c · k outgoing edges. Ito and Iwama [ACM Trans. Algorithms, to appear] gave an algorithm to enumerate all cisolated maximal cliques in O(4 c ·c 4 · E) time. We extend this to enumerating all maximal ci ..."
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In an undirected graph G = (V,E), a set of k vertices is called cisolated if it has less than c · k outgoing edges. Ito and Iwama [ACM Trans. Algorithms, to appear] gave an algorithm to enumerate all cisolated maximal cliques in O(4 c ·c 4 · E) time. We extend this to enumerating all maximal cisolated cliques (which are a superset) and improve the running time bound to O(2.89 c ·c 2 ·E), using modifications which also facilitate parallelizing the enumeration. Moreover, we introduce a more restricted and a more general isolation concept and show that both lead to faster enumeration algorithms. Finally, we extend our considerations to splexes (a relaxation of the clique notion), providing a W[1]hardness result when the size of the splex is the parameter and a fixedparameter algorithm for enumerating isolated splexes when the parameter describes the degree of isolation.
Editing Graphs into Disjoint Unions of Dense Clusters
"... Abstract. In the ΠCluster 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 grap ..."
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Abstract. In the ΠCluster 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 wellstudied 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 (sdefective cliques), 2) having average degree at least VK  − s (averagesplexes), and 3) having average degree at least µ · (VK  − 1) (µcliques), where s and µ are a fixed integer and a fixed rational number, respectively. We first show that the ΠCluster Editing problem is NPcomplete for all three density measures. Then, we study the fixedparameter tractability of the three clustering problems, showing that the first two problems are fixedparameter 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. 1
Finding dense subgraphs of sparse graphs
 Proc. IPEC 2012 2012
"... Abstract. We investigate the computational complexity of the DensestkSubgraph (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 complexi ..."
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Abstract. We investigate the computational complexity of the DensestkSubgraph (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 fixedparameter tractable with respect to either of the parameters maximum degree and hindex 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. 1
Inequalities for the Number of Walks in Graphs
"... We investigate the growth of the number wk of walks of length k in undirected graphs as well as related inequalities. In the first part, we derive the inequalities w2a+c · w2(a+b)+c ≤ w2a · w2(a+b+c) and w2a+c(v, v) · w2(a+b)+c(v, v) ≤ w2a(v, v) · w2(a+b+c)(v, v) for the number wk(v, v) of closed ..."
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We investigate the growth of the number wk of walks of length k in undirected graphs as well as related inequalities. In the first part, we derive the inequalities w2a+c · w2(a+b)+c ≤ w2a · w2(a+b+c) and w2a+c(v, v) · w2(a+b)+c(v, v) ≤ w2a(v, v) · w2(a+b+c)(v, v) for the number wk(v, v) of closed walks of length k starting at a given vertex v. The first is a direct implication of a matrix inequality by Marcus and Newman and generalizes two inequalities by Lagarias et al. and Dress & Gutman. We then use an inequality of Blakley and Dixon to show the inequality wk 2ℓ+p ≤ w2ℓ+pk · w k−1 2ℓ which also generalizes the inequality by Dress
1 Detecting dense communities in large social and information networks with the
, 1210
"... Detecting and characterizing dense subgraphs (tight communities) in social and information networks is an important exploratory tool in social network analysis. Several approaches have been proposed that either (i) partition the whole network into ”clusters”, even in low density region, or (ii) ar ..."
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Detecting and characterizing dense subgraphs (tight communities) in social and information networks is an important exploratory tool in social network analysis. Several approaches have been proposed that either (i) partition the whole network into ”clusters”, even in low density region, or (ii) are aimed at finding a single densest community (and need to be iterated to find the next one). As social networks grow larger both approaches (i) and (ii) result in algorithms too slow to be practical, in particular when speed in analyzing the data is required. In this paper we propose an approach that aims at balancing efficiency of computation and expressiveness/manageability of the output community representation. We define the notion of a partial dense cover (PDC) of a graph. Intuitively a PDC of a graph is a collection of sets of nodes that (a) each set forms a disjoint dense induced subgraphs and (b) its removal leaves the residual graph without dense regions. Exact computation of PDC is an NPcomplete problem, thus, we propose an efficient heuristic algorithms for computing a PDC which we christen Core & Peel. Moreover we propose a novel benchmarking technique that allows us to evaluate algorithms for computing PDC using the classical IR concepts of precision and recall even without a golden standard. Tests on 25 social and technological networks from the Stanford Large Network Dataset Collection confirm that Core & Peel is efficient and attains very high precison and recall. 1.
A More Complicated Hardness Proof for Finding Densest Subgraphs in Bounded Degree Graphs
, 2014
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Spectral Radius, and the Energy of Graphs Inequalities for the Number of Walks, the Spectral Radius, and the Energy of Graphs
"... We unify and generalize several inequalities for the number wk of walks of length k in graphs. The new inequalities use an arbitrary nonnegative weighting of the vertices. This allows an application of the theorems to subsets of vertices, i.e., these inequalities consider the number wk(S, S) of walk ..."
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We unify and generalize several inequalities for the number wk of walks of length k in graphs. The new inequalities use an arbitrary nonnegative weighting of the vertices. This allows an application of the theorems to subsets of vertices, i.e., these inequalities consider the number wk(S, S) of walks of length k that start at a vertex of a given vertex subset S and that end within the same subset. In particular, we show a weighted sandwich theorem for Hermitian matrices which generalizes a theorem by Marcus and Newman and which implies w2a+c(S, S) · w2a+2b+c(S, S) ≤ w2a(S, S) · w2(a+b+c)(S, S), a unification and generalization of inequalities by Lagarias et al. and by Dress & Gutman. Further, we show a theorem for nonnegative symmetric matrices that implies w2`+p(S, S) k ≤ w2`(S, S)k−1 · w2`+pk(S, S), which is a unification and generalization of inequalities by Erdős & Simonovits, by Dress & Gutman, and by Ilic ́ & Stevanović. Both results can be translated into corresponding forms for matrix or graph densities. In the end, these results are used to generalize lower bounds for the spectral radius and upper bounds for the energy of graphs. 1