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, and as well, by the vertex cover number. Finally, we give for both problems polynomial-time algorithms for some subclasses of acyclic digraphs.

Relation Between Broadcast Domination and Multipacking Numbers on Chordal Graphs

“…However, polynomial-time algorithms are known for trees and more generally, strongly chordal graphs [4]. See [10] for other references concerning algorithmic results on the two problems.…”

“…However, polynomial-time algorithms are known for trees and more generally, strongly chordal graphs [10]. See [11] for other references concerning algorithmic results on the two problems. We know that mp(G) ≤ γ b (G), since broadcast domination and multipacking are dual problems [12], and γ b (G) ≤ rad(G) [5], where rad(G) is the radius of G. Moreover, it is known that γ b (G) ≤ 2 mp(G) + 3 [13] and it is a conjecture that γ b (G) ≤ 2 mp(G) for every graph G [13].…”