Scalable Approximation Algorithm for Network Immunization

The problem of identifying important players in a given network is of pivotal importance for viral marketing, public health management, network security and various other fields of social network analysis. In this work we find the most important vertices in a graph G = (V;E) to immunize so as the chances of an epidemic outbreak is minimized. This problem is directly relevant to minimizing the impact of a contagion spread (e.g. flu virus, computer virus and rumor) in a graph (e.g. social network, computer network) with a limited budget (e.g. the number of available vaccines, antivirus software, filters). It is well known that this problem is computationally intractable (it is NP-hard). In this work we reformulate the problem as a budgeted combinational optimization problem and use techniques from spectral graph theory to design an efficient greedy algorithm to find a subset of vertices to be immunized. We show that our algorithm takes less time compared to the state of the art algorithm. Thus our algorithm is scalable to networks of much larger sizes than best known solutions proposed earlier. We also give analytical bounds on the quality of our algorithm. Furthermore, we evaluate the efficacy of our algorithm on a number of real world networks and demonstrate that the empirical performance of algorithm supplements the theoretical bounds we present, both in terms of approximation guarantees and computational efficiency

Connected searching of weighted trees

AbstractIn this paper we consider the problem of connected edge searching of weighted trees. Barrière et al. claim in [L. Barrière, P. Flocchini, P. Fraigniaud, N. Santoro, Capture of an intruder by mobile agents, in: SPAA’02: Proceedings of the Fourteenth Annual ACM Symposium on Parallel Algorithms and Architectures, ACM, New York, NY, USA, 2002, pp. 200–209] that there exists a polynomial-time algorithm for finding an optimal search strategy, that is, a strategy that minimizes the number of used searchers. However, due to some flaws in their algorithm, the problem turns out to be open. It is proven in this paper that the considered problem is strongly NP-complete even for node-weighted trees (the weight of each edge is 1) with one vertex of degree greater than 2. It is also shown that there exists a polynomial-time algorithm for finding an optimal connected search strategy for a given bounded degree tree with arbitrary weights on the edges and on the vertices. This is an FPT algorithm with respect to the maximum degree of a tree

Decontamination of Hypercubes by Mobile Agents

In this article we consider the decontamination problem in a hypercube network of size n. The nodes of the network are assumed to be contaminated and they have to be decontaminated by a sufficient number of agents. An agent is a mobile entity that asynchronously moves along the network links and decontaminates all the nodes it touches. A decontaminated node that is not occupied by an agent is re-contaminated if it has a contaminated neighbor. We consider some variations of the model based on the capabilities of mobile agents: locality, where the agents can only access local information; visibility, where they can “see” the state of their neighbors; and cloning, where they can create copies of them- selves. We also consider synchronicity as an alternative system requirement. For each model, we design a decontamination strategy and we make several observations. For agents with locality, our strategy is based on the use of a coordinator that leads the other agents. Our strategy results in an optimal number of agents, Theta( n/√ log n ), and requires O(n log n) moves and O(n log n) time steps. For agents with visibility, we assume that the agents can move autonomously. In this setting, our decontamination strategy achieves an optimal time complexity (log n time steps), but the number of agents increases to n . Finally, we show that when the agents have the capability to clone combined with either visibility or synchronicity, we can reduce the move complexity—which becomes optimal— at the expense of an increase in the number of agents