Broadcast Independence and Broadcast Packing are FPT parameterized by treewidth plus diameter via DP on nice tree decompositions, W[1]-hard for pathwidth, and admit a constant-factor approximation parameterized by treewidth.
On the broadcast independence number of caterpillars
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On the parameterized complexity of Broadcast Independence and Broadcast Packing
Broadcast Independence and Broadcast Packing are FPT parameterized by treewidth plus diameter via DP on nice tree decompositions, W[1]-hard for pathwidth, and admit a constant-factor approximation parameterized by treewidth.