Complexity of Eliminating (Majority) Illusion in Directed Networks

We study illusion elimination problems on directed social networks where each vertex is colored either red or blue. A vertex is under majority illusion if it has more red out-neighbors than blue out-neighbors when there are more blue vertices than red ones in the network. In a more general phenomenon of $p$-illusion, a vertex under illusion has at least $p$ fraction of the out-neighbors (as opposed to $1/2$ for majority) of a vertex is red. In the directed illusion elimination problem, we recolor minimum number of vertices so that no vertex is under $p$-illusion, for $p\in (0,1)$. Unfortunately, the problem is NP-hard for $p =1/2$ even when the network is a grid. Moreover, the problem is NP-hard and W[2]-hard when parameterized by the number of recoloring for each $p \in (0,1)$ even on bipartite DAGs. Thus, we can neither get a polynomial time algorithm on DAGs, unless P=NP, nor we can get a FPT algorithm even by combining solution size and directed graph parameters that measure distance from acyclicity, unless FPT=W[2]. Furthermore, the problem is NP-complete in graphs of degree at most three for $p\in (0,\frac{2}{3}]$. We show that the problem can be solved in polynomial time in structured, sparse networks such as outerplanar networks, outward grids, trees, and cycles. Finally, we show tractable algorithms parameterized by treewidth of the underlying undirected graph and maximum deficiency of the graph combined, and by the number of vertices under illusion.