Centrality metrics are among the main tools in social network analysis. Being central for a user of a network leads to several benefits to the user: central users are highly influential and play key roles within the network. Therefore, the optimization problem of increasing the centrality of a network user recently received considerable attention. Given a network and a target user $v$, the centrality maximization problem consists in creating $k$ new links incident to $v$ in such a way that the centrality of $v$ is maximized, according to some centrality metric. Most of the algorithms proposed in the literature are based on showing that a given centrality metric is monotone and submodular with respect to link addition. However, this property does not hold for several shortest-path based centrality metrics if the links are undirected. In this paper we study the centrality maximization problem in undirected networks for one of the most important shortest-path based centrality measures, the coverage centrality. We provide several hardness and approximation results. We first show that the problem cannot be approximated within a factor greater than $1-1/e$, unless $P=NP$, and, under the stronger gap-ETH hypothesis, the problem cannot be approximated within a factor better than $1/n^{o(1)}$, where $n$ is the number of users. We then propose two greedy approximation algorithms, and show that, by suitably combining them, we can guarantee an approximation factor of $Ω(1/\sqrt{n})$. We experimentally compare the solutions provided by our approximation algorithm with optimal solutions computed by means of an exact IP formulation. We show that our algorithm produces solutions that are very close to the optimum.
Betweenness centrality is a fundamental centrality measure in social network analysis. Given a large-scale network, how can we find the most central nodes? This question is of key importance to numerous important applications that rely on betweenness centrality, including community detection and understanding graph vulnerability. Despite the large amount of work on designing scalable approximation algorithms for betweenness centrality, estimating it on large-scale networks remains a computational challenge. In this paper, we study the Betweenness Centrality Maximization problem: given a graph …
The study of vertex centrality measures is a key aspect of network analysis. Naturally, such centrality measures have been generalized to groups of vertices; for popular measures it was shown that the problem of finding the most central group is $\mathcal{NP}$-hard. As a result, approximation algorithms to maximize group centralities were introduced recently. Despite a nearly-linear running time, approximation algorithms for group betweenness and (to a lesser extent) group closeness are rather slow on large networks due to high constant overheads. That is why we introduce GED-Walk centrality, …
We study the popular centrality measure known as effective conductance or in some circles as information centrality. This is an important notion of centrality for undirected networks, with many applications, e.g., for random walks, electrical resistor networks, epidemic spreading, etc. In this paper, we first reinterpret this measure in terms of modulus (energy) of families of walks on the network. This modulus centrality measure coincides with the effective conductance measure on simple undirected networks, and extends it to much more general situations, e.g., directed networks as well. Secon…
We consider the influence maximization problem (selecting $k$ seeds in a network maximizing the expected total influence) on undirected graphs under the linear threshold model. On the one hand, we prove that the greedy algorithm always achieves a $(1 - (1 - 1/k)^k + Ω(1/k^3))$-approximation, showing that the greedy algorithm does slightly better on undirected graphs than the generic $(1- (1 - 1/k)^k)$ bound which also applies to directed graphs. On the other hand, we show that substantial improvement on this bound is impossible by presenting an example where the greedy algorithm can obtain at …
In network analysis and graph mining, closeness centrality is a popular measure to infer the importance of a vertex. Computing closeness efficiently for individual vertices received considerable attention. The NP-hard problem of group closeness maximization, in turn, is more challenging: the objective is to find a vertex group that is central as a whole and state-of-the-art heuristics for it do not scale to very big graphs yet. In this paper, we present new local search heuristics for group closeness maximization. By using randomized approximation techniques and dynamic data structures, our al…
Centrality measures characterize important nodes in networks. Efficiently computing such nodes has received a lot of attention. When considering the generalization of computing central groups of nodes, challenging optimization problems occur. In this work, we study two such problems, group-harmonic maximization and group-closeness maximization both from a theoretical and from an algorithm engineering perspective. On the theoretical side, we obtain the following results. For group-harmonic maximization, unless $P=NP$, there is no polynomial-time algorithm that achieves an approximation factor b…
In this paper we are interested in a version of the All-pairs Shortest Paths problem (APSP) that fits neither in the exact nor in the approximate case. We define a measure of centrality of a shortest path, related to the ``importance'' of such shortest path in the graph, and propose an algorithm based on the idea of progressive sampling that, for {\it any fixed constants} $0 < ε$, $ δ< 1$, given an undirected graph $G$ with non-negative edge weights, outputs with probability $1 - δ$ a data structure of size $n \cdot \textrm{Diam}_V(G)$, where $\textrm{Diam}_V(G)$ is the vertex diameter of $G$,…
We present KADABRA, a new algorithm to approximate betweenness centrality in directed and undirected graphs, which significantly outperforms all previous approaches on real-world complex networks. The efficiency of the new algorithm relies on two new theoretical contributions, of independent interest. The first contribution focuses on sampling shortest paths, a subroutine used by most algorithms that approximate betweenness centrality. We show that, on realistic random graph models, we can perform this task in time $|E|^{\frac{1}{2}+o(1)}$ with high probability, obtaining a significant speedup…
The modularity is a quality function in community detection, which was introduced by Newman and Girvan (2004). Community detection in graphs is now often conducted through modularity maximization: given an undirected graph $G=(V,E)$, we are asked to find a partition $\mathcal{C}$ of $V$ that maximizes the modularity. Although numerous algorithms have been developed to date, most of them have no theoretical approximation guarantee. Recently, to overcome this issue, the design of modularity maximization algorithms with provable approximation guarantees has attracted significant attention in the …
The betweenness centrality (BC) of a node in a network (or graph) is a measure of its importance in the network. BC is widely used in a large number of environments such as social networks, transport networks, security/mobile networks and more. We present an O(n)-round distributed algorithm for computing BC of every vertex as well as all pairs shortest paths (APSP) in a directed unweighted network, where n is the number of vertices and m is the number of edges. We also present O(n)-round distributed algorithms for computing APSP and BC in a weighted directed acyclic graph (dag). Our algorithms…
In this paper we provide an algorithm which given any $m$-edge $n$-vertex directed graph with integer capacities at most $U$ computes a maximum $s$-$t$ flow for any vertices $s$ and $t$ in $m^{11/8+o(1)}U^{1/4}$ time with high probability. This running time improves upon the previous best of $\tilde{O}(m^{10/7} U^{1/7})$ (Mądry 2016), $\tilde{O}(m \sqrt{n} \log U)$ (Lee Sidford 2014), and $O(mn)$ (Orlin 2013) when the graph is not too dense or has large capacities. We achieve this result by leveraging recent advances in solving undirected flow problems on graphs. We show that in the maximum fl…
In this paper we study a maximization version of the classical Edge Dominating Set (EDS) problem, namely, the Upper EDS problem, in the realm of Parameterized Complexity. In this problem, given an undirected graph $G$, a positive integer $k$, the question is to check whether $G$ has a minimal edge dominating set of size at least $k$. We obtain the following results for Upper EDS. We prove that Upper EDS admits a kernel with at most $4k^2-2$ vertices. We also design a fixed-parameter tractable (FPT) algorithm for Upper EDS running in time $2^{\mathcal{O}(k)} \cdot n^{\mathcal{O}(1)}$.
Graphical models are a rich language for describing high-dimensional distributions in terms of their dependence structure. While there are algorithms with provable guarantees for learning undirected graphical models in a variety of settings, there has been much less progress in the important scenario when there are latent variables. Here we study Restricted Boltzmann Machines (or RBMs), which are a popular model with wide-ranging applications in dimensionality reduction, collaborative filtering, topic modeling, feature extraction and deep learning. The main message of our paper is a strong dic…
We consider a variant of the prize collecting Steiner tree problem in which we are given a \emph{directed graph} $D=(V,A)$, a monotone submodular prize function $p:2^V \rightarrow \mathbb{R}^+ \cup \{0\}$, a cost function $c:V \rightarrow \mathbb{Z}^{+}$, a root vertex $r \in V$, and a budget $B$. The aim is to find an out-subtree $T$ of $D$ rooted at $r$ that costs at most $B$ and maximizes the prize function. We call this problem \emph{Directed Rooted Submodular Tree} (\textbf{DRSO}). Very recently, Ghuge and Nagarajan [SODA\ 2020] gave an optimal quasi-polynomial-time $O\left(\frac{\log n'}…
Closeness is a widely-used centrality measure in social network analysis. For a node it indicates the reciprocal of the average shortest-path distance to the other nodes of the network. While the identification of the k nodes with highest closeness received significant attention, many applications are actually interested in finding a group of nodes that is central as a whole. For this problem, only recently a greedy algorithm has been proposed [Chen et al., ADC 2016]. The approximation factor of (1 - 1/e) proposed by Chen et al. for this algorithm does not hold, though, as we show in this vers…
Given a weighted hypergraph $\mathcal{H}(V, \mathcal{E} \subseteq 2^V, w)$, the approximate $k$-cover problem seeks for a size-$k$ subset of $V$ that has the maximum weighted coverage by \emph{sampling only a few hyperedges} in $\mathcal{E}$. The problem has emerged from several network analysis applications including viral marketing, centrality maximization, and landmark selection. Despite many efforts, even the best approaches require $O(k n \log n)$ space complexities, thus, cannot scale to, nowadays, humongous networks without sacrificing formal guarantees. In this paper, we propose BCA, a…
In the $k$-Steiner Orientation problem, we are given a mixed graph, that is, with both directed and undirected edges, and a set of $k$ terminal pairs. The goal is to find an orientation of the undirected edges that maximizes the number of terminal pairs for which there is a path from the source to the sink. The problem is known to be W[1]-hard when parameterized by k and hard to approximate up to some constant for FPT algorithms assuming Gap-ETH. On the other hand, no approximation factor better than $O(k)$ is known. We show that $k$-Steiner Orientation is unlikely to admit an approximation al…
For the General Factor problem we are given an undirected graph $G$ and for each vertex $v\in V(G)$ a finite set $B_v$ of non-negative integers. The task is to decide if there is a subset $S\subseteq E(G)$ such that $deg_S(v)\in B_v$ for all vertices $v$ of $G$. The maxgap of a finite integer set $B$ is the largest $d\ge 0$ such that there is an $a\ge 0$ with $[a,a+d+1]\cap B=\{a,a+d+1\}$. Cornuéjols (1988) showed that if the maxgap of all sets $B_v$ is at most 1, then the decision version of General Factor is poly-time solvable. Dudycz and Paluch (2018) extended this result for the minimizati…
We study a constrained shortest path problem in group-labeled graphs with nonnegative edge length, called the shortest non-zero path problem. Depending on the group in question, this problem includes two types of tractable variants in undirected graphs: one is the parity-constrained shortest path/cycle problem, and the other is computing a shortest noncontractible cycle in surface-embedded graphs. For the shortest non-zero path problem with respect to finite abelian groups, Kobayashi and Toyooka (2017) proposed a randomized, pseudopolynomial-time algorithm via permanent computation. For a slig…
