Dominance Maximization in Uncertain Graphs

Abstract

An uncertain graph is defined as a collection of nodes with their interconnected edges, where each edge is assigned with its existence probability value. Motivated by many real-world scenarios, in this paper, we define the notion of dominance for a given subset of nodes as the sum of its cardinality and the total probability with which the remaining nodes in the graph are dominated by the subset. For a given uncertain graph $\mathcal {G}(\mathcal {V}, \mathcal {E}, \mathcal {P})$, and a positive integer k, we introduce the Problem of Dominance Maximization where the goal is to choose a k-element subset to maximize the dominance in the uncertain graph. First, we show that the dominance function is non-negative, monotone, and submodular. Subsequently, we show that the problem of maximizing dominance in an uncertain graph is NP-hard. Then we propose an incremental greedy strategy based on the marginal gain in dominance computation that leads to $(1-\frac{1}{e})$-factor approximate solution. With a slight change, we show that this strategy can be made efficient without losing the approximation ratio. Finally, we exploit the submodularity property of the dominance function and construct a combinatorial object called ‘Pruned Submodularity Graph’ and use this effectively to solve the dominance maximization problem. All the methodologies have been analyzed to understand their time and space requirements. We implement the proposed solution approaches with real-world network datasets and conduct many experiments. We observe that for larger datasets the Pruned Submodularity Graph-based approach takes reasonable computational time and produces better solutions in terms of dominance value compared to the baseline methods.