On the Complexity of Broadcast Domination and Multipacking in Digraphs

Abstract

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[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. Finally, we give for both problems polynomial-time algorithms for some subclasses of acyclic digraphs.

2-limited broadcast domination on grid graphs

“…In general, the complexity of deciding whether a given graph G has a multipacking number of at least a given integer B is unknown. There does however exist a (2 + o(1)) approximation algorithm for this decision problem on undirected graphs [2] and the problem was recently shown to be NP-complete for digraphs [18]. The dual of the LP relaxation of the k-limited broadcast domination ILP is fractional k-limited multipacking.…”

“…Broadcast domination has recently been extended to digraphs [18]. A vertex x hears a broadcast of strength s from a vertex y if the directed distance from y to x is at most s. The problem of deciding whether there exists a dominating broadcast of cost at most a given integer B is NP-complete [18].…”

On the Complexity of Broadcast Domination and Multipacking in Digraphs

“…We thank the four referees for their careful reviewing. In particular, thanks to the referee who pointed out that Lemma 8 from the conference version [19] is wrong: this has led to the corrected version of Theorem 27. Florent Foucaud is partially funded by the IFCAM project "Applications of graph homomorphisms" (MA/IFCAM/18/39) and by the ANR project HOSIGRA (ANR-17-CE40-0022).…”

“…Note that in the conference version of this paper[19], we have claimed the same algorithm for the maximum out-degree, instead of the maximum degree. However, this algorithm was based on an incorrect claim (Lemma 8 in[19]). …”