A broadcast on a connected graph is a function f that assigns each vertex v an integer f(v) with 0 <= f(v) <= ecc(v) where ecc(v) denotes the eccentricity of v. A vertex u hears a broadcasting vertex v (with f(v)>0) if u is at distance at most f(v) from v. Beyond the classical broadcast domination problem, where every vertex is required to hear at least one vertex, two variants raise intriguing combinatorial and algorithmic questions. In an independent broadcast, no broadcasting vertex hears another broadcasting vertex, while a broadcast packing requires that every vertex hears at most one broadcasting vertex. The corresponding problems Broadcast Independence and Broadcast Packing ask for broadcasts of values at least k under these constraints, where the value is the sum of the broadcast values. We initiate a systematic study of the parameterized complexity of such problems. We prove that Broadcast Independence and Broadcast Packing are FPT parameterized by the treewidth plus the diameter of G, with a family of dynamic-programming algorithms over nice tree decompositions. We obtain as a corollary that both problems are FPT parameterized by k and the treewidth of G and XP for treewidth only. The latter result shows that the known algorithm for trees (Bessy and Rautenbach, DAM 2022) can indeed be extended to bounded treewidth graphs. On the negative side, we show that Broadcast Independence is W[1]-hard parameterized by the pathwidth of G. Note that this result completes the picture for parameter k and treewidth for Broadcast Independence since it is known to be W[1]-hard for k only. We complement these results by showing that a weighted version of both problems, where the input comes with a weight function on the edges, is W[1]-hard parameterized by the vertex cover of G. Finally, we provide a constant-factor approximation algorithm parameterized by treewidth for Broadcast Independence.

翻译：在连通图中，广播是一个函数 \( f \)，它为每个顶点 \( v \) 分配一个整数 \( f(v) \) 满足 \( 0 \leq f(v) \leq ecc(v) \)，其中 \( ecc(v) \) 表示顶点 \( v \) 的离心率。若顶点 \( u \) 与广播顶点 \( v \)（满足 \( f(v) > 0 \)）的距离不超过 \( f(v) \)，则称 \( u \) 能听到 \( v \) 的广播。除经典的广播支配问题（要求每个顶点至少听到一个广播）外，两种变体引发了有趣的组合与算法问题。在独立广播中，没有广播顶点能听到另一个广播顶点；而广播打包则要求每个顶点至多听到一个广播顶点。对应问题【广播独立性】与【广播打包】要求在这些约束下广播值（即所有广播值的总和）至少为 \( k \)。我们首次系统研究了此类问题的参数化复杂性。我们证明：若以图 \( G \) 的树宽加直径为参数，【广播独立性】与【广播打包】属于 FPT，并基于优美树分解设计了一族动态规划算法。作为推论，这两个问题关于参数 \( k \) 与图 \( G \) 的树宽属于 FPT，且仅关于树宽属于 XP。后者表明已知的树算法（Bessy and Rautenbach, DAM 2022）确实可扩展至有界树宽图。在否定方面，我们证明【广播独立性】关于图 \( G \) 的路径宽属于 W[1]-困难。注意：该结果完善了【广播独立性】在参数 \( k \) 与树宽下的图景——已知该问题仅关于参数 \( k \) 即属于 W[1]-困难。此外，我们证明这两个问题的加权版本（输入附带边权函数）关于图 \( G \) 的顶点覆盖属于 W[1]-困难。最后，我们针对【广播独立性】在树宽参数化下给出了一个常数因子近似算法。