Pervasive Domination

Abstract

Inspired by the implicit or explicit persuasion scenario, which characterizes social media platforms, we analyze a novel domination problem named Pervasive Partial Domination (PPD). We consider a social network modeled by a digraph $G=(V,E)$ where an arc $(u,v)\in E$ represents the capability of an individual u to persuade an individual v. We are looking for a set $S \subset V$ of social change individuals, of minimum cost, who combined enable to reach the desired behavior. The impact of S is measured by a set function f(S) that is the sum of the degree of belief of all the individuals in the network and p is the desired target. We show that the natural greedy algorithm, for the PPD problem, provides an approximation guarantee, $\left( \ln \frac{ p-f(\emptyset )}{\beta }+2\right)$ where $\beta >0$ represents the minimum gain on the function f one can attain by bribing an additional individual when the target p is (almost) reached. The proposed solution can be generalized to the weighted partial sumbmodular cover problem providing a better approximation with respect to the state of the art.