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, in Leszek Gąsieniec, Ralf Klasing, Tomasz Radzik, Combinatorial Algorithms  31st International Workshop, Springer, p. 264276. 10.1007/9783030489663_20
Type
Communication / ConférenceExternal document link
https://hal.archivesouvertes.fr/hal02793880Date
2020Conference title
Combinatorial Algorithms, 31st International Workshop, IWOCA 2020Conference date
2020Conference country
FRANCEBook title
Combinatorial Algorithms  31st International WorkshopBook author
Leszek Gąsieniec, Ralf Klasing, Tomasz RadzikPublisher
Springer
ISBN
9783030489663
Pages
264276
Publication identifier
Metadata
Show full item recordAbstract (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 outdegree. Finally, we give for both problems polynomialtime algorithms for some subclasses of acyclic digraphs.Subjects / Keywords
Broadcast domination; Dominating set; Multipacking; Directed graphs; Parameterized complexityRelated items
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