The Complexity of Finding Harmless Individuals in Social Networks
Résumé
In this paper, we introduce a domination-related problem called Harmless Set: given a graph G=(V,E)G=(V,E), a threshold function t:V→Nt:V→N and an integer kk, find a subset of vertices V′⊆VV′⊆V of size at least kk such that every vertex vv in VV has less than t(v)t(v) neighbors in V′V′. We study its parameterized complexity and the approximation of the associated maximization problem. When the parameter is kk, we show that the problem is W[2]-complete in general and W[1]-complete if all thresholds are bounded by a constant. Moreover, we prove that, if P≠NPP≠NP, the maximization version is not View the MathML sourcen12−ε-approximable for any ε>0ε>0 even when all thresholds are at most two. When each threshold is equal to the degree of the vertex, we show that Harmless Set is fixed-parameter tractable for parameter kk and the maximization version is APX-complete. We give a polynomial-time algorithm for graphs of bounded treewidth and a polynomial-time approximation scheme for planar graphs. Finally, we show that the parametric dual problem (n−k)(n−k)-Harmless Set is fixed-parameter tractable for a large family of threshold functions.