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Register a new userTop-k Connected Overlapping Densest Subgraphs in Dual Networks
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Added by
Mohammad Mehdi Hosseinzadeh
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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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.
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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.
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.
In the minimum $k$-edge-connected spanning subgraph ($k$-ECSS) problem the goal is to find the minimum weight subgraph resistant to up to $k-1$ edge failures. This is a central problem in network design, and a natural generalization of the minimum spanning tree (MST) problem. While the MST problem has been studied extensively by the distributed computing community, for $k geq 2$ less is known in the distributed setting. In this paper, we present fast randomized distributed approximation algorithms for $k$-ECSS in the CONGEST model. Our first contribution is an $widetilde{O}(D + sqrt{n})$-round $O(log{n})$-approximation for 2-ECSS, for a graph with $n$ vertices and diameter $D$. The time complexity of our algorithm is almost tight and almost matches the time complexity of the MST problem. For larger constant values of $k$ we give an $widetilde{O}(n)$-round $O(log{n})$-approximation. Additionally, in the special case of unweighted 3-ECSS we show how to improve the time complexity to $O(D log^3{n})$ rounds. All our results significantly improve the time complexity of previous algorithms.
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.
In the Survivable Network Design Problem (SNDP), the input is an edge-weighted (di)graph $G$ and an integer $r_{uv}$ for every pair of vertices $u,vin V(G)$. The objective is to construct a subgraph $H$ of minimum weight which contains $r_{uv}$ edge-disjoint (or node-disjoint) $u$-$v$ paths. This is a fundamental problem in combinatorial optimization that captures numerous well-studied problems in graph theory and graph algorithms. In this paper, we consider the version of the problem where we are given a $lambda$-edge connected (di)graph $G$ with a non-negative weight function $w$ on the edges and an integer $k$, and the objective is to find a minimum weight spanning subgraph $H$ that is also $lambda$-edge connected, and has at least $k$ fewer edges than $G$. In other words, we are asked to compute a maximum weight subset of edges, of cardinality up to $k$, which may be safely deleted from $G$. Motivated by this question, we investigate the connectivity properties of $lambda$-edge connected (di)graphs and obtain algorithmically significant structural results. We demonstrate the importance of our structural results by presenting an algorithm running in time $2^{O(k log k)} |V(G)|^{O(1)}$ for $lambda$-ECS, thus proving its fixed-parameter tractability. We follow up on this result and obtain the {em first polynomial compression} for $lambda$-ECS on unweighted graphs. As a consequence, we also obtain the first fixed parameter tractable algorithm, and a polynomial kernel for a parameterized version of the classic Mininum Equivalent Graph problem. We believe that our structural results are of independent interest and will play a crucial role in the design of algorithms for connectivity-constrained problems in general and the SNDP problem in particular.
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