On the Complexity of Broadcast Domination and Multipacking in Digraphs
Foucaud, Florent; Gras, Benjamin; Perez, Anthony; Sikora, Florian (2020), On the Complexity of Broadcast Domination and Multipacking in Digraphs, Algorithmica, 83, p. 2651–2677. 10.1007/s00453021008285
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Article accepté pour publication ou publiéDate
2020Journal name
AlgorithmicaVolume
83Publisher
Springer
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2651–2677
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Foucaud, FlorentGras, Benjamin
Perez, Anthony
Sikora, Florian
Laboratoire d'analyse et modélisation de systèmes pour l'aide à la décision [LAMSADE]
Abstract (EN)
We study the complexity of the two dual covering and packing distancebased problems Broadcast Domination and Multipacking in digraphs. A dominating broadcast of a digraph D is a function f:V(D)→N such that for each vertex v of D, there exists a vertex t with f(t)>0 having a directed path to v of length at most f(t). The cost of f is the sum of f(v) over all vertices v. A multipacking is a set S of vertices of D such that for each vertex v of D and for every integer d, there are at most d vertices from S within directed distance at most d from v. The maximum size of a multipacking of D is a lower bound to the minimum cost of a dominating broadcast of D. Let Broadcast Domination denote the problem of deciding whether a given digraph D has a dominating broadcast of cost at most k, and Multipacking the problem of deciding whether D has a multipacking of size at least k. It is known that Broadcast Domination is polynomialtime solvable for the class of all undirected graphs (that is, symmetric digraphs), while polynomialtime algorithms for Multipacking are known only for a few classes of undirected graphs. We prove that Broadcast Domination and Multipacking are both NPcomplete for digraphs, even for planar layered acyclic digraphs of small maximum degree. Moreover, when parameterized by the solution cost/solution size, we show that the problems are respectively W[2]hard and W[1]hard. We also show that Broadcast Domination is FPT on acyclic digraphs, and that it does not admit a polynomial kernel for such inputs, unless the polynomial hierarchy collapses to its third level. In addition, we show that both problems are FPT when parameterized by the solution cost/solution size together with the maximum (out)degree, and as well, by the vertex cover number. Finally, we give for both problems polynomialtime algorithms for some subclasses of acyclic digraphs.Subjects / Keywords
Broadcast Domination; digraphsRelated items
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