We study illusion elimination problems on directed social networks where each vertex is colored either red or blue. A vertex is under \textit{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, 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 recolorings 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]. 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 by the number of vertices under illusion.

翻译：我们研究有向社交网络中的幻觉消除问题，其中每个顶点被染成红色或蓝色。当网络中蓝色顶点多于红色顶点时，若某顶点的红色出邻居数量超过蓝色出邻居数量，则该顶点处于"多数幻觉"之下。在更一般的$p$-幻觉现象中，顶点至少$p$比例的出邻居（而非多数幻觉的$1/2$）为红色。在有向幻觉消除问题中，我们需重新染色最少顶点数量，使得对于$p\in(0,1)$，没有顶点处于$p$-幻觉状态。不幸的是，当网络为网格图时，该问题在$p=1/2$时即为NP难问题。此外，对于每个$p\in(0,1)$，即使在二分有向无环图上，该问题也是NP难且W[2]-难的。因此，我们既无法在有向无环图上获得多项式时间算法（除非P=NP），也无法通过结合解规模与度量偏离无环性程度的有向图参数获得FPT算法（除非FPT=W[2]）。我们证明该问题可在结构化稀疏网络（如外平面图、外向网格、树和环）上多项式时间求解。最后，我们展示以底层无向图的树宽和幻觉顶点数为参数的可解算法。