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Register a new userComputing the k Densest Subgraphs of a Graph
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Added by
Riccardo Dondi
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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Computing cohesive subgraphs is a central problem in graph theory. While many formulations of cohesive subgraphs lead to NP-hard problems, finding a densest subgraph can be done in polynomial time. As such, the densest subgraph model has emerged as the most popular notion of cohesiveness. Recently, the data mining community has started looking into the problem of computing k densest subgraphs in a given graph, rather than one, with various restrictions on the possible overlap between the subgraphs. However, there seems to be very little known on this important and natural generalization from a theoretical perspective. In this paper we hope to remedy this situation by analyzing three natural variants of the k densest subgraphs problem. Each variant differs depending on the amount of overlap that is allowed between the subgraphs. In one extreme, when no overlap is allowed, we prove that the problem is NP-hard for k >= 3. On the other extreme, when overlap is allowed without any restrictions and the solution subgraphs only have to be distinct, we show that the problem is fixed-parameter tractable with respect to k, and admits a PTAS for constant k. Finally, when a limited of overlap is allowed between the subgraphs, we prove that the problem is NP-hard for k = 2.
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A central problem in graph mining is finding dense subgraphs, with several applications in different fields, a notable example being identifying communities. While a lot of effort has been put on the problem of finding a single dense subgraph, only recently the focus has been shifted to the problem of finding a set of densest subgraphs. Some approaches aim at finding disjoint subgraphs, while in many real-world networks communities are often overlapping. An approach introduced to find possible overlapping subgraphs is the Top-k Overlapping Densest Subgraphs problem. For a given integer k >= 1, the goal of this problem is to find a set of k densest subgraphs that may share some vertices. The objective function to be maximized takes into account both the density of the subgraphs and the distance between subgraphs in the solution. The Top-k Overlapping Densest Subgraphs problem has been shown to admit a 1/10-factor approximation algorithm. Furthermore, the computational complexity of the problem has been left open. In this paper, we present contributions concerning the approximability and the computational complexity of the problem. For the approximability, we present approximation algorithms that improves the approximation factor to 1/2 , when k is bounded by the vertex set, and to 2/3 when k is a constant. For the computational complexity, we show that the problem is NP-hard even when k = 3.
The Densest $k$-Subgraph (D$k$S) problem, and its corresponding minimization problem Smallest $p$-Edge Subgraph (S$p$ES), have come to play a central role in approximation algorithms. This is due both to their practical importance, and their usefulness as a tool for solving and establishing approximation bounds for other problems. These two problems are not well understood, and it is widely believed that they do not an admit a subpolynomial approximation ratio (although the best known hardness results do not rule this out). In this paper we generalize both D$k$S and S$p$ES from graphs to hypergraphs. We consider the Densest $k$-Subhypergraph problem (given a hypergraph $(V, E)$, find a subset $Wsubseteq V$ of $k$ vertices so as to maximize the number of hyperedges contained in $W$) and define the Minimum $p$-Union problem (given a hypergraph, choose $p$ of the hyperedges so as to minimize the number of vertices in their union). We focus in particular on the case where all hyperedges have size 3, as this is the simplest non-graph setting. For this case we provide an $O(n^{4(4-sqrt{3})/13 + epsilon}) leq O(n^{0.697831+epsilon})$-approximation (for arbitrary constant $epsilon > 0$) for Densest $k$-Subhypergraph and an $tilde O(n^{2/5})$-approximation for Minimum $p$-Union. We also give an $O(sqrt{m})$-approximation for Minimum $p$-Union in general hypergraphs. Finally, we examine the interesting special case of interval hypergraphs (instances where the vertices are a subset of the natural numbers and the hyperedges are intervals of the line) and prove that both problems admit an exact polynomial time solution on these instances.
Networks are largely used for modelling and analysing data and relations among them. Recently, it has been shown that the use of a single network may not be the optimal choice, since a single network may misses some aspects. Consequently, it has been proposed to use a pair of networks to better model all the aspects, and the main approach is referred to as dual networks (DNs). DNs are two related graphs (one weighted, the other unweighted) that share the same set of vertices and two different edge sets. In DNs is often interesting to extract common subgraphs among the two networks that are maximally dense in the conceptual network and connected in the physical one. The simplest instance of this problem is finding a common densest connected subgraph (DCS), while we here focus on the detection of the Top-k Densest Connected subgraphs, i.e. a set k subgraphs having the largest density in the conceptual network which are also connected in the physical network. We formalise the problem and then we propose a heuristic to find a solution, since the problem is computationally hard. A set of experiments on synthetic and real networks is also presented to support our approach.
We investigate the parameterized complexity of finding subgraphs with hereditary properties on graphs belonging to a hereditary graph class. Given a graph $G$, a non-trivial hereditary property $Pi$ and an integer parameter $k$, the general problem $P(G,Pi,k)$ asks whether there exists $k$ vertices of $G$ that induce a subgraph satisfying property $Pi$. This problem, $P(G,Pi,k)$ has been proved to be NP-complete by Lewis and Yannakakis. The parameterized complexity of this problem is shown to be W[1]-complete by Khot and Raman, if $Pi$ includes all trivial graphs but not all complete graphs and vice versa; and is fixed-parameter tractable (FPT), otherwise. As the problem is W[1]-complete on general graphs when $Pi$ includes all trivial graphs but not all complete graphs and vice versa, it is natural to further investigate the problem on restricted graph classes. Motivated by this line of research, we study the problem on graphs which also belong to a hereditary graph class and establish a framework which settles the parameterized complexity of the problem for various hereditary graph classes. In particular, we show that: $P(G,Pi,k)$ is solvable in polynomial time when the graph $G$ is co-bipartite and $Pi$ is the property of being planar, bipartite or triangle-free (or vice-versa). $P(G,Pi,k)$ is FPT when the graph $G$ is planar, bipartite or triangle-free and $Pi$ is the property of being planar, bipartite or triangle-free, or graph $G$ is co-bipartite and $Pi$ is the property of being co-bipartite. $P(G,Pi,k)$ is W[1]-complete when the graph $G$ is $C_4$-free, $K_{1,4}$-free or a unit disk graph and $Pi$ is the property of being either planar or bipartite.
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