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 most a $(1- (1 - 1/k)^k + O(1/k^{0.2}))$ approximation. This result stands in contrast to the previous work on the independent cascade model. Like the linear threshold model, the greedy algorithm obtains a $(1-(1-1/k)^k)$-approximation on directed graphs in the independent cascade model. However, Khanna and Lucier showed that, in undirected graphs, the greedy algorithm performs substantially better: a $(1-(1-1/k)^k + c)$ approximation for constant $c > 0$. Our results show that, surprisingly, no such improvement occurs in the linear threshold model. Finally, we show that, under the linear threshold model, the approximation ratio $(1 - (1 - 1/k)^k)$ is tight if 1) the graph is directed or 2) the vertices are weighted. In other words, under either of these two settings, the greedy algorithm cannot achieve a $(1 - (1 - 1/k)^k + f(k))$-approximation for any positive function $f(k)$. The result in setting 2) is again in a sharp contrast to Khanna and Lucier's $(1 - (1 - 1/k)^k + c)$-approximation result for the independent cascade model, where the $(1 - (1 - 1/k)^k + c)$ approximation guarantee can be extended to the setting where vertices are weighted. We also discuss extensions to more generalized settings including those with edge-weighted graphs.
In graph theory, the longest path problem is the problem of finding a simple path of maximum length in a given graph. For some small classes of graphs, the problem can be solved in polynomial time [2, 4], but it remains NP-hard on general graphs, since it includes the Hamiltonian path problem as a special case [3]. Motivated by finding a simple, quick algorithm for finding long paths in large graphs, in this paper we show a greedy algorithm with a time complexity of O(n^2 (n+m)), where n is the number of the vertices and m is the number of edges.
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…
We study the problem of deterministically exploring an undirected and initially unknown graph with $n$ vertices either by a single agent equipped with a set of pebbles, or by a set of collaborating agents. The vertices of the graph are unlabeled and cannot be distinguished by the agents, but the edges incident to a vertex have locally distinct labels. The graph is explored when all vertices have been visited by at least one agent. In this setting, it is known that for a single agent without pebbles $Θ(\log n)$ bits of memory are necessary and sufficient to explore any graph with at most $n$ ve…
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 …
Using a connected dominating set (CDS) to serve as the virtual backbone of a wireless sensor network (WSN) is an effective way to save energy and reduce the impact of broadcasting storms. Since nodes may fail due to accidental damage or energy depletion, it is desirable that the virtual backbone is fault tolerant. This could be modeled as a k-connected, m-fold dominating set ((k,m)-CDS). Given a virtual undirected network G=(V,E), a subset C\subset V is a (k,m)-CDS of G if (i) G[C], the subgraph of G induced by C is k-connected, and (ii) each node in V\C has at least m neighbors in C. We prese…
Our motivation is to improve on the best approximation guarantee known for the problem of finding a minimum-cost 2-node connected spanning subgraph of a given undirected graph with nonnegative edge costs. We present an LP (Linear Programming) relaxation based on partition constraints. The special case where the input contains a spanning tree of zero cost is called 2NC-TAP. We present a greedy algorithm for 2NC-TAP, and we analyze it via dual-fitting for our partition LP relaxation. Keywords: 2-node connected graphs, approximation algorithms, connectivity augmentation, greedy algorithm, network…
In this paper we present the first deterministic polynomial time algorithm for determining the existence of a Hamiltonian cycle and finding a Hamiltonian cycle in general graphs. Our algorithm can also solve the Hamiltonian path problem in the traceable graphs. The space complexity of our algorithm is O(n^4). The time complexity are theoretically O(n^5*d^2) on average and O(n^6*d^2) in the worst case respectively, where d is the maximum degree of vertex. With parallel computing, the space complexity can be improved to O(n^3) and the time complexity to O(n^3*d^2) on average and O(n^4*d^2) in th…
We study the problem of maximizing the number of spanning trees in a connected graph by adding at most $k$ edges from a given candidate edge set. We give both algorithmic and hardness results for this problem: - We give a greedy algorithm that, using submodularity, obtains an approximation ratio of $(1 - 1/e - ε)$ in the exponent of the number of spanning trees for any $ε> 0$ in time $\tilde{O}(m ε^{-1} + (n + q) ε^{-3})$, where $m$ and $q$ is the number of edges in the original graph and the candidate edge set, respectively. Our running time is optimal with respect to the input size up to log…
Moving an autonomous agent through an unknown environment is one of the crucial problems for robotics and network analysis. Therefore, it received a lot of attention in the last decades and was analyzed in many different settings. The graph exploration problem is a theoretical and abstract model, where an algorithm has to decide how the agent, also called explorer, moves through a network such that every point of interest is visited at least once. For its decisions, the knowledge of the algorithm is limited by the perception of the explorer. There are different models regarding the perception …
We present the first non-trivial online algorithms for the non-uniform, multicommodity buy-at-bulk (MC-BB) network design problem in undirected and directed graphs. Our competitive ratios qualitatively match the best known approximation factors for the corresponding offline problems. The main engine for our results is an online reduction theorem of MC-BB problems to their single-sink (SS-BB) counterparts. We use the concept of junction-tree solutions (Chekuri et al., FOCS 2006) that play an important role in solving the offline versions of the problem via a greedy subroutine -- an inherently o…
In a model of network communication based on a random walk in an undirected graph, what subset of nodes (subject to constraints on the set size), enable the fastest spread of information? The dynamics of spread is described by a process dual to the movement from informed to uninformed nodes. In this setting, an optimal set $A$ minimizes the sum of the expected first hitting times $F(A)$, of random walks that start at nodes outside the set. In this paper,the problem is reformulated so that the search for solutions is restricted to a class of optimal and "near" optimal subsets of the graph. We i…
This paper defines the toroidal small world labeling problem that asks for a labeling of the vertices of a network such that the labels possess information that allows a compact routing scheme in the network. We consider the problem over a small world network model we propose. Both the model and the compact routing scheme have applications in peer-to-peer networks. The proposed model is based on the model of Kleinberg (2000), and generates an undirected two-dimensional torus with one random long-range edge per vertex. These random edges create forbidden cycles that mimic the underlying torus t…
Missouri Univ. of Science and Technology（密苏里科技大学）；University of California, Merced（加州大学默塞德分校）；Sapienza University of Rome（罗马第一大学）；University of Perugia（佩鲁贾大学）
In this paper, we study the Orienteering Aisle-graphs Single-access Problem (OASP), a variant of the orienteering problem for a robot moving in a so-called single-access aisle-graph, i.e., a graph consisting of a set of rows that can be accessed from one side only. Aisle-graphs model, among others, vineyards or warehouses. Each aisle-graph vertex is associated with a reward that a robot obtains when visits the vertex itself. As the robot's energy is limited, only a subset of vertices can be visited with a fully charged battery. The objective is to maximize the total reward collected by the rob…
In the classical Node-Disjoint Paths (NDP) problem, the input consists of an undirected $n$-vertex graph $G$, and a collection $\mathcal{M}=\{(s_1,t_1),\ldots,(s_k,t_k)\}$ of pairs of its vertices, called source-destination, or demand, pairs. The goal is to route the largest possible number of the demand pairs via node-disjoint paths. The best current approximation for the problem is achieved by a simple greedy algorithm, whose approximation factor is $O(\sqrt n)$, while the best current negative result is an $Ω(\log^{1/2-δ}n)$-hardness of approximation for any constant $δ$, under standard com…
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…
Let $G = (V,E,w)$ be a weighted undirected graph on $|V| = n$ vertices and $|E| = m$ edges, let $k \ge 1$ be any integer, and let $ε< 1$ be any parameter. We present the following results on fast constructions of spanners with near-optimal sparsity and lightness, which culminate a long line of work in this area. (By near-optimal we mean optimal under Erdős' girth conjecture and disregarding the $ε$-dependencies.) - There are (deterministic) algorithms for constructing $(2k-1)(1+ε)$-spanners for $G$ with a near-optimal sparsity of $O(n^{1/k} \log(1/ε)/ε))$. The first algorithm can be implemente…
We study the classical Node-Disjoint Paths (NDP) problem: given an undirected $n$-vertex graph G, together with a set {(s_1,t_1),...,(s_k,t_k)} of pairs of its vertices, called source-destination, or demand pairs, find a maximum-cardinality set of mutually node-disjoint paths that connect the demand pairs. The best current approximation for the problem is achieved by a simple greedy $O(\sqrt{n})$-approximation algorithm. A special case of the problem called NDP-Grid, where the underlying graph is a grid, has been studied extensively. The best current approximation algorithm for NDP-Grid achiev…
