Approximation algorithms for NP-complete problems on planar graphs
Journal of the ACM (JACM)
Fixed-parameter tractability and completeness II: on completeness for W[1]
Theoretical Computer Science
A partial k-arboretum of graphs with bounded treewidth
Theoretical Computer Science
On Some Tighter Inapproximability Results (Extended Abstract)
ICAL '99 Proceedings of the 26th International Colloquium on Automata, Languages and Programming
Maximizing the spread of influence through a social network
Proceedings of the ninth ACM SIGKDD international conference on Knowledge discovery and data mining
The turing way to parameterized complexity
Journal of Computer and System Sciences - Special issue on Parameterized computation and complexity
On the approximability of influence in social networks
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
Approximation Algorithms and Hardness for Domination with Propagation
APPROX '07/RANDOM '07 Proceedings of the 10th International Workshop on Approximation and the 11th International Workshop on Randomization, and Combinatorial Optimization. Algorithms and Techniques
Blocking links to minimize contamination spread in a social network
ACM Transactions on Knowledge Discovery from Data (TKDD)
Discrete Applied Mathematics
Resource minimization for fire containment
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Parameterized complexity of generalized domination problems
Discrete Applied Mathematics
Treewidth governs the complexity of target set selection
Discrete Optimization
Parameterized Complexity
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In this paper, we introduce the Robust Set problem: given a graph G=(V,E), a threshold function t:V→N and an integer k, find a subset of vertices V′⊆V of size at least k such that every vertex v in G has less than t(v) neighbors in V′. This problem occurs in the context of the spread of undesirable agents through a network (virus, ideas, fire, …). Informally speaking, the problem asks to find the largest subset of vertices with the property that if anything bad happens in it then this will have no consequences on the remaining graph. The threshold t(v) of a vertex v represents its reliability regarding its neighborhood; that is, how many neighbors can be infected before v gets himself infected. We study in this paper the parameterized complexity of Robust Set and the approximation of the associated maximization problem. When the parameter is k, we show that this problem is W[2]-complete in general and W[1]-complete if all thresholds are constant bounded. Moreover, we prove that, if P≠NP, the maximization version is not n1−ε- approximable for any ε0 even when all thresholds are at most two. When each threshold is equal to the degree of the vertex, we show that k-Robust Set is fixed-parameter tractable for parameter k and the maximization version is APX-complete. We give a polynomial-time algorithm for graphs of bounded treewidth and a PTAS for planar graphs. Finally, we show that the parametric dual problem (n−k)-Robust Set is fixed-parameter tractable for a large family of threshold functions.