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 not be robust to vertex/edge failure. Indeed, a densest subgraph may not be well-connected, which implies that the subgraph may be disconnected by removing only a few vertices/edges within it. In this paper, we provide an algorithmic framework to find a dense subgraph that is well-connected in terms of vertex/edge connectivity. Specifically, we introduce the following problems: given a graph $G=(V,E,w)$ and a positive integer/real $k$, we are asked to find $S\subseteq V$ that maximizes the density $d(S)$ under the constraint that $G[S]$ is $k$-vertex/edge-connected. For both problems, we propose polynomial-time (bicriteria and ordinary) approximation algorithms, using classic Mader's theorem in graph theory and its extensions.
Many graph mining applications rely on detecting subgraphs which are near-cliques. There exists a dichotomy between the results in the existing work related to this problem: on the one hand the densest subgraph problem (DSP) which maximizes the average degree over all subgraphs is solvable in polynomial time but for many networks fails to find subgraphs which are near-cliques. On the other hand, formulations that are geared towards finding near-cliques are NP-hard and frequently inapproximable due to connections with the Maximum Clique problem. In this work, we propose a formulation which comb…
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.
Dense subgraph discovery is an important primitive in graph mining, which has a wide variety of applications in diverse domains. In the densest subgraph problem, given an undirected graph $G=(V,E)$ with an edge-weight vector $w=(w_e)_{e\in E}$, we aim to find $S\subseteq V$ that maximizes the density, i.e., $w(S)/|S|$, where $w(S)$ is the sum of the weights of the edges in the subgraph induced by $S$. Although the densest subgraph problem is one of the most well-studied optimization problems for dense subgraph discovery, there is an implicit strong assumption; it is assumed that the weights of…
In the densest subgraph problem, given an edge-weighted undirected graph $G=(V,E,w)$, we are asked to find $S\subseteq V$ that maximizes the density, i.e., $w(S)/|S|$, where $w(S)$ is the sum of weights of the edges in the subgraph induced by $S$. This problem has often been employed in a wide variety of graph mining applications. However, the problem has a drawback; it may happen that the obtained subset is too large or too small in comparison with the size desired in the application at hand. In this study, we address the size issue of the densest subgraph problem by generalizing the density …
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)…
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…
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 consider two optimization problems related to finding dense subgraphs. The densest at-least-k-subgraph problem (DalkS) is to find an induced subgraph of highest average degree among all subgraphs with at least k vertices, and the densest at-most-k-subgraph problem (DamkS) is defined similarly. These problems are related to the well-known densest k-subgraph problem (DkS), which is to find the densest subgraph on exactly k vertices. We show that DalkS can be approximated efficiently, while DamkS is nearly as hard to approximate as the densest k-subgraph problem.
\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 $G$.
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…
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…
Recently [Bhattacharya et al., STOC 2015] provide the first non-trivial algorithm for the densest subgraph problem in the streaming model with additions and deletions to its edges, i.e., for dynamic graph streams. They present a $(0.5-ε)$-approximation algorithm using $\tilde{O}(n)$ space, where factors of $ε$ and $\log(n)$ are suppressed in the $\tilde{O}$ notation. However, the update time of this algorithm is large. To remedy this, they also provide a $(0.25-ε)$-approximation algorithm using $\tilde{O}(n)$ space with update time $\tilde{O}(1)$. In this paper we improve the algorithms by Bha…
Given a large graph, the densest-subgraph problem asks to find a subgraph with maximum average degree. When considering the top-$k$ version of this problem, a naïve solution is to iteratively find the densest subgraph and remove it in each iteration. However, such a solution is impractical due to high processing cost. The problem is further complicated when dealing with dynamic graphs, since adding or removing an edge requires re-running the algorithm. In this paper, we study the top-$k$ densest-subgraph problem in the sliding-window model and propose an efficient fully-dynamic algorithm. The …
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…
Given a connected graph $G$ on $n$ vertices and a positive integer $k\le n$, a subgraph of $G$ on $k$ vertices is called a $k$-subgraph in $G$. We design combinatorial approximation algorithms for finding a connected $k$-subgraph in $G$ such that its density is at least a factor $Ω(\max\{n^{-2/5},k^2/n^2\})$ of the density of the densest $k$-subgraph in $G$ (which is not necessarily connected). These particularly provide the first non-trivial approximations for the densest connected $k$-subgraph problem on general graphs.
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…
In the Densest k-Subgraph problem, given a graph G and a parameter k, one needs to find a subgraph of G induced on k vertices that contains the largest number of edges. There is a significant gap between the best known upper and lower bounds for this problem. It is NP-hard, and does not have a PTAS unless NP has subexponential time algorithms. On the other hand, the current best known algorithm of Feige, Kortsarz and Peleg, gives an approximation ratio of n^(1/3-epsilon) for some specific epsilon > 0 (estimated at around 1/60). We present an algorithm that for every epsilon > 0 approximates th…
Given a task $\mathcal{T}$, a set of experts $V$ with multiple skills and a social network $G(V, W)$ reflecting the compatibility among the experts, team formation is the problem of identifying a team $C \subseteq V$ that is both competent in performing the task $\mathcal{T}$ and compatible in working together. Existing methods for this problem make too restrictive assumptions and thus cannot model practical scenarios. The goal of this paper is to consider the team formation problem in a realistic setting and present a novel formulation based on densest subgraphs. Our formulation allows modeli…
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…
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…
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 …
In this paper, we consider the problem of approximating the densest subgraph in the dynamic graph stream model. In this model of computation, the input graph is defined by an arbitrary sequence of edge insertions and deletions and the goal is to analyze properties of the resulting graph given memory that is sub-linear in the size of the stream. We present a single-pass algorithm that returns a $(1+ε)$ approximation of the maximum density with high probability; the algorithm uses $O(ε^{-2} n \polylog n)$ space, processes each stream update in $\polylog (n)$ time, and uses $\poly(n)$ post-proces…
The densest k-subgraph (DkS) problem (i.e. find a size k subgraph with maximum number of edges), is one of the notorious problems in approximation algorithms. There is a significant gap between known upper and lower bounds for DkS: the current best algorithm gives an ~ O(n^{1/4}) approximation, while even showing a small constant factor hardness requires significantly stronger assumptions than P != NP. In addition to interest in designing better algorithms, a number of recent results have exploited the conjectured hardness of densest k-subgraph and its variants. Thus, understanding the approxi…
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…
We consider the problem of identifying a team of skilled individuals for collaboration, in the presence of a social network. Each node in the social network may be an expert in one or more skills. Edge weights specify affinity or collaborative compatibility between respective nodes. Given a project that requires a set of specified number of skilled individuals in each area of expertise, the goal is to identify a team that maximizes the collaborative compatibility. For example, the requirement may be to form a team that has at least three databases experts and at least two theory experts. We ex…
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 …
In distributed networks, it is often useful for the nodes to be aware of dense subgraphs, e.g., such a dense subgraph could reveal dense subtructures in otherwise sparse graphs (e.g. the World Wide Web or social networks); these might reveal community clusters or dense regions for possibly maintaining good communication infrastructure. In this work, we address the problem of self-awareness of nodes in a dynamic network with regards to graph density, i.e., we give distributed algorithms for maintaining dense subgraphs that the member nodes are aware of. The only knowledge that the nodes need is…
Robustness is a critical measure of the resilience of large networked systems, such as transportation and communication networks. Most prior works focus on the global robustness of a given graph at large, e.g., by measuring its overall vulnerability to external attacks or random failures. In this paper, we turn attention to local robustness and pose a novel problem in the lines of subgraph mining: given a large graph, how can we find its most robust local subgraph (RLS)? We define a robust subgraph as a subset of nodes with high communicability among them, and formulate the RLS-PROBLEM of find…
While in many graph mining applications it is crucial to handle a stream of updates efficiently in terms of {\em both} time and space, not much was known about achieving such type of algorithm. In this paper we study this issue for a problem which lies at the core of many graph mining applications called {\em densest subgraph problem}. We develop an algorithm that achieves time- and space-efficiency for this problem simultaneously. It is one of the first of its kind for graph problems to the best of our knowledge. In a graph $G = (V, E)$, the "density" of a subgraph induced by a subset of node…
