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. 264-276. 10.1007/978-3-030-48966-3_20
TypeCommunication / Conférence
External document linkhttps://hal.archives-ouvertes.fr/hal-02793880
Conference titleCombinatorial Algorithms, 31st International Workshop, IWOCA 2020
Book titleCombinatorial Algorithms - 31st International Workshop
Book authorLeszek Gąsieniec, Ralf Klasing, Tomasz Radzik
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Abstract (EN)We study the complexity of the two dual covering and packing distance-based 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 polynomial-time solvable for the class of all undirected graphs (that is, symmetric digraphs), while polynomial-time algorithms for Multipacking are known only for a few classes of undirected graphs. We prove that Broadcast Domination and Multipacking are both NP-complete 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-hard and W-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. Finally, we give for both problems polynomial-time algorithms for some subclasses of acyclic digraphs.
Subjects / KeywordsBroadcast domination; Dominating set; Multipacking; Directed graphs; Parameterized complexity
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