# Balancing spreads of influence in a social network

## TL;DR

This paper generalizes the problem of balancing influence spreads in social networks to multiple campaigns, providing approximation algorithms and hardness results for various settings, highlighting computational limits under certain complexity assumptions.

## Contribution

It extends previous models to multiple campaigns, offering new approximation algorithms and establishing hardness results under Gap-ETH and one-way function assumptions.

## Key findings

- Provided approximation algorithms for correlated and heterogeneous settings with multiple campaigns.
- Proved hardness of approximation within certain factors under Gap-ETH and one-way function assumptions.
- Extended the understanding of computational limits in influence balancing problems.

## Abstract

The personalization of our news consumption on social media has a tendency to reinforce our pre-existing beliefs instead of balancing our opinions. This finding is a concern for the health of our democracies which rely on an access to information providing diverse viewpoints. To tackle this issue from a computational perspective, Garimella et al. (NIPS'17) modeled the spread of these viewpoints, also called campaigns, using the well-known independent cascade model and studied an optimization problem that aims at balancing information exposure in a social network when two opposing campaigns propagate in the network. The objective in their $NP$-hard optimization problem is to maximize the number of people that are exposed to either both or none of the viewpoints. For two different settings, one corresponding to a model where campaigns spread in a correlated manner, and a second one, where the two campaigns spread in a heterogeneous manner, they provide constant ratio approximation algorithms. In this paper, we investigate a more general formulation of this problem. That is, we assume that $\mu$ different campaigns propagate in a social network and we aim to maximize the number of people that are exposed to either $\nu$ or none of the campaigns, where $\mu\ge\nu\ge2$. We provide dedicated approximation algorithms for both the correlated and heterogeneous settings. Interestingly, for the heterogeneous setting with $\nu\ge 3$, we give a reduction leading to several approximation hardness results. Maybe most importantly, we obtain that the problem cannot be approximated within a factor of $n^{-g(n)}$ for any $g(n)=o(1)$ assuming Gap-ETH, denoting with $n$ the number of nodes in the social network. For $\nu \ge 4$, there is no $n^{-\epsilon}$-approximation algorithm if a certain class of one-way functions exists, where $\epsilon > 0$ is a given constant which depends on $\nu$.