Parameterized Approximability of Maximizing the Spread of Influence in Networks

TL;DR

This paper investigates the computational complexity of influence maximization in social networks, revealing strong inapproximability results and fixed-parameter tractability under certain conditions, thus advancing understanding of influence spread optimization.

Contribution

It establishes new hardness and approximability bounds for influence maximization with thresholds, including inapproximability in polynomial time and fixed-parameter tractability in bounded degree graphs.

Findings

Strong inapproximability in FPT-time for general thresholds

Polynomial inapproximability when thresholds equal degrees

FPT algorithms exist for bounded degree graphs

Abstract

In this paper, we consider the problem of maximizing the spread of influence through a social network. Given a graph with a threshold value~$thr(v)$ attached to each vertex~$v$, the spread of influence is modeled as follows: A vertex~$v$ becomes "active" (influenced) if at least $thr(v)$ of its neighbors are active. In the corresponding optimization problem the objective is then to find a fixed number of vertices to activate such that the number of activated vertices at the end of the propagation process is maximum. We show that this problem is strongly inapproximable in fpt-time with respect to (w.r.t.) parameter $k$ even for very restrictive thresholds. In the case that the threshold of each vertex equals its degree, we prove that the problem is inapproximable in polynomial time and it becomes $r(n)$-approximable in fpt-time w.r.t. parameter $k$ for any strictly increasing function $r$. Moreover, we show that the decision version is W[1]-hard w.r.t. parameter $k$ but becomes fixed-parameter tractable on bounded degree graphs.