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 \(f:V(D)\rightarrow {\mathbb {N}}\) 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.

我们研究有向图中两个基于双重覆盖和打包距离的问题的复杂性，即广播支配和多重打包。甲支配广播一个有向图的d是一个函数\（F：V（d）\ RIGHTARROW {\ mathbb {N}} \），使得对于每一个顶点v的d，存在顶点吨与\（F（T） > 0 \）到v的定向路径的长度最多为f（t）。f的代价是所有顶点v的f（v）的总和。一个multipacking是一组小号的顶点d，使得对于每个顶点v的d和用于每个整数d，最多有d从顶点小号定向距离内至多d从v。的multipacking的最大尺寸d是低级绑定到的一个主导广播的最小成本d。让Broadcast Domination表示确定给定的有向图D是否具有最大成本为k的主导广播的问题，而Multipacking表示确定是否有成本为k的主导广播的问题。D具有大小至少为k的多重包装。众所周知，广播控制对于所有无向图（即对称有向图）的类都是多项式时间可解的，而用于Multipacking的多项式时间算法仅对于少数几类无向图是已知的。我们证明，对于有向图，即使对于最大程度较小的平面分层无环有向图，广播控制和多重打包也都是NP完全的。此外，当用解决方案成本/解决方案大小进行参数化时，我们表明问题分别是W [2] -hard和W [1] -hard。我们还展示了广播统治非循环图上的FPT是FPT，并且除非多项式层次结构崩溃到其第三级，否则它不允许此类输入使用多项式内核。另外，我们表明，当通过解决方案成本/解决方案大小以及最大（向外）度以及顶点覆盖数对参数进行参数化时，这两个问题都是FPT。最后，针对这两个问题，我们给出了非循环有向图的某些子类的多项式时间算法。