Computational Complexity of Competitive Diffusion on (Un)weighted Graphs

TL;DR

This paper studies the computational complexity of finding pure Nash equilibria in competitive diffusion games on various types of graphs, revealing hardness results and identifying cases where the problem is solvable efficiently.

Contribution

It provides new complexity results for the existence of pure Nash equilibria in competitive diffusion on different graph classes, including hardness and polynomial-time solvable cases.

Findings

NP-hardness on series-parallel graphs with weights

W[1]-hardness parameterized by number of players

Polynomial-time algorithms for certain graph classes

Abstract

Consider an undirected graph modeling a social network, where the vertices represent users, and the edges do connections among them. In the competitive diffusion game, each of a number of players chooses a vertex as a seed to propagate his/her opinion, and then it spreads along the edges in the graphs. The objective of every player is to maximize the number of vertices the opinion infects. In this paper, we investigate a computational problem of asking whether a pure Nash equilibrium exists in the competitive diffusion game on unweighed and weighted graphs, and present several negative and positive results. We first prove that the problem is W[1]-hard when parameterized by the number of players even for unweighted graphs. We also show that the problem is NP-hard even for series-parallel graphs with positive integer weights, and is NP-hard even for forests with arbitrary integer weights. Furthermore, we show that the problem for forest of paths with arbitrary weights is solvable in pseudo-polynomial time; and it is solvable in quadratic time if a given graph is unweighted. We also prove that the problem for chain, cochain, and threshold graphs with arbitrary integer weights is solvable in polynomial time.