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.
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,…
We give an exact $O(nk^2)$ algorithm for finding the densest k subgraph in outerplanar graphs. We extend this to an exact $O(nk^2 8^b)$ algorithm for finding the densest k subgraph in b-outerplanar graphs. Finally, we hypothesize that Baker's PTAS technique will not work for the densest k subgraph problem in planar graphs.
We give the first fully dynamic algorithm which maintains a $(1-ε)$-approximate densest subgraph in worst-case time $\text{poly}(\log n, ε^{-1})$ per update. Dense subgraph discovery is an important primitive for many real-world applications such as community detection, link spam detection, distance query indexing, and computational biology. We approach the densest subgraph problem by framing its dual as a graph orientation problem, which we solve using an augmenting path-like adjustment technique. Our result improves upon the previous best approximation factor of $\left(\frac{1}{4} - ε\right)…
\textsc{Densest $k$-Subgraph} is the problem to find a vertex subset $S$ of size $k$ such that the number of edges in the subgraph induced by $S$ is maximized. In this paper, we show that \textsc{Densest $k$-Subgraph} is fixed parameter tractable when parameterized by neighborhood diversity, block deletion number, distance-hereditary deletion number, and cograph deletion number, respectively. Furthermore, we give a $2$-approximation $2^{\tc(G)/2}n^{O(1)}$-time algorithm where $\tc(G)$ is the twin cover number of an input graph $G$.
Dense subgraph discovery is an important graph-mining primitive with a variety of real-world applications. One of the most well-studied optimization problems for dense subgraph discovery is the densest subgraph problem, where given an edge-weighted undirected graph $G=(V,E,w)$, we are asked to find $S\subseteq V$ that maximizes the density $d(S)$, i.e., half the weighted average degree of the induced subgraph $G[S]$. This problem can be solved exactly in polynomial time and well-approximately in almost linear time. However, a densest subgraph has a structural drawback, namely, the subgraph may…
Given a graph and an integer $k$, Densest $k$-Subgraph is the algorithmic task of finding the subgraph on $k$ vertices with the maximum number of edges. This is a fundamental problem that has been subject to intense study for decades, with applications spanning a wide variety of fields. The state-of-the-art algorithm is an $O(n^{1/4 + ε})$-factor approximation (for any $ε> 0$) due to Bhaskara et al. [STOC '10]. Moreover, the so-called log-density framework predicts that this is optimal, i.e. it is impossible for an efficient algorithm to achieve an $O(n^{1/4 - ε})$-factor approximation. In the…
In this paper we study the problem of discovering a timeline of events in a temporal network. We model events as dense subgraphs that occur within intervals of network activity. We formulate the event-discovery task as an optimization problem, where we search for a partition of the network timeline into k non-overlapping intervals, such that the intervals span subgraphs with maximum total density. The output is a sequence of dense subgraphs along with corresponding time intervals, capturing the most interesting events during the network lifetime. A naive solution to our optimization problem ha…
Given an undirected graph $G$, the Densest $k$-subgraph problem (DkS) asks to compute a set $S \subset V$ of cardinality $\left\lvert S\right\rvert \leq k$ such that the weight of edges inside $S$ is maximized. This is a fundamental NP-hard problem whose approximability, inspite of many decades of research, is yet to be settled. The current best known approximation algorithm due to Bhaskara et al. (2010) computes a $\mathcal{O}\left({n^{1/4 + ε}}\right)$ approximation in time $n^{\mathcal{O}\left(1/ε\right)}$, for any $ε> 0$. We ask what are some "easier" instances of this problem? We propose …
Given a graph, the densest subgraph problem asks for a set of vertices such that the average degree among these vertices is maximized. Densest subgraph has numerous applications in learning, e.g., community detection in social networks, link spam detection, correlation mining, bioinformatics, and so on. Although there are efficient algorithms that output either exact or approximate solutions to the densest subgraph problem, existing algorithms may violate the privacy of the individuals in the network, e.g., leaking the existence/non-existence of edges. In this paper, we study the densest subgr…
Query Complexity of Densest Sub(hyper)graph Problems
Pacific Northwest National Laboratory（太平洋西北国家实验室）；School of Computing, University of Utah（犹他大学计算机学院）；Department of Computer Science and Engineering, Texas A&M University（德克萨斯农工大学计算机科学工程系）
We propose a new conjecture on hardness of low-degree $2$-CSP's, and show that new hardness of approximation results for Densest $k$-Subgraph and several other problems, including a graph partitioning problem, and a variation of the Graph Crossing Number problem, follow from this conjecture. The conjecture can be viewed as occupying a middle ground between the $d$-to-$1$ conjecture, and hardness results for $2$-CSP's that can be obtained via standard techniques, such as Parallel Repetition combined with standard $2$-prover protocols for the 3SAT problem. We hope that this work will motivate fu…
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 max…
Finding dense subgraphs of a large graph is a standard problem in graph mining that has been studied extensively both for its theoretical richness and its many practical applications. In this paper we introduce a new family of dense subgraph objectives, parameterized by a single parameter $p$, based on computing generalized means of degree sequences of a subgraph. Our objective captures both the standard densest subgraph problem and the maximum $k$-core as special cases, and provides a way to interpolate between and extrapolate beyond these two objectives when searching for other notions of de…
Differentially private algorithms allow large-scale data analytics while preserving user privacy. Designing such algorithms for graph data is gaining importance with the growth of large networks that model various (sensitive) relationships between individuals. While there exists a rich history of important literature in this space, to the best of our knowledge, no results formalize a relationship between certain parallel and distributed graph algorithms and differentially private graph analysis. In this paper, we define \emph{locally adjustable} graph algorithms and show that algorithms of thi…
We propose a simple and computationally efficient method for dense subgraph discovery, which is a classic problem both in theory and in practice. It is well known that dense subgraphs can have strong correlation with structures of interest in real-world networks across various domains ranging from social networks to biological systems [Gionis and Tsourakakis `15]. For the densest subgraph problem, Charikar's greedy algorithm [Asashiro `00, Charikar `00] is very simple, and can typically find result of quality much better than the provable factor 2-approximation, which makes it very popular in …
We consider the problem of maintaining a $(1-ε)$-approximation to the densest subgraph (DSG) in an undirected multigraph as it undergoes edge insertions and deletions (the fully dynamic setting). Sawlani and Wang [SW20] developed a data structure that, for any given $ε> 0$, maintains a $(1-ε)$-approximation with $O(\log^4 n/ε^6)$ worst-case update time for edge operations, and $O(1)$ query time for reporting the density value. Their data structure was the first to achieve near-optimal approximation, and improved previous work that maintained a $(1/4 - ε)$ approximation in amortized polylogarit…
The densest subgraph problem (DSG) aiming at finding an induced subgraph such that the average edge-weights of the subgraph is maximized, is a well-studied problem. However, when the input graph is a hypergraph, the existing notion of DSG fails to capture the fact that a hyperedge partially belonging to an induced sub-hypergraph is also a part of the sub-hypergraph. To resolve the issue, we suggest a function $f_e:\mathbb{Z}_{\ge0}\rightarrow \mathbb{R}_{\ge 0}$ to represent the partial edge-weight of a hyperedge $e$ in the input hypergraph $\mathcal{H}=(V,\mathcal{E},f)$ and formulate a gener…
In the densest subgraph problem, given a weighted undirected graph $G(V,E,w)$, with non-negative edge weights, we are asked to find a subset of nodes $S\subseteq V$ that maximizes the degree density $w(S)/|S|$, where $w(S)$ is the sum of the edge weights induced by $S$. This problem is a well studied problem, known as the {\em densest subgraph problem}, and is solvable in polynomial time. But what happens when the edge weights are negative? Is the problem still solvable in polynomial time? Also, why should we care about the densest subgraph problem in the presence of negative weights? In this …
