Faster Parameterized Broadcasting

Given a connected graph $G$ and a source $s \in V(G)$, what is the smallest number of rounds necessary for all vertices of $G$ to receive a message initially only held by $s$, where at each round every informed vertex passes the message to one of its neighbors? This problem is called Telephone Broadcast and is suprisingly hard: it remains NP-hard on cycles intersecting at a single shared vertex, in particular, graphs of pathwidth 2 with a linear feedback vertex set of size 1, as well as on graphs with treedepth at most 6 [Egami et al.; MFCS '25]. Vertex cover number, vertex integrity, and distance to clique are among the few parameters for which Telephone Broadcast is fixed-parameter tractable. There is a $2^{\mathcal{O}(\mathrm{vc}^3)} n^{\mathcal{O}(1)}$-time algorithm parameterized by vertex cover number $\mathrm{vc}$ [Fomin, Fraigniaud, Golovach; TCS '24], a double-exponential algorithm parameterized by vertex integrity $\mathrm{vi}$, and a $2^{\mathcal{O}(k^2)} n^{\mathcal{O}(1)}$-time algorithm parameterized by distance to clique $k$ [Egami et al.; MFCS '25]. In this paper, we give improved parameterized algorithms for Telephone Broadcast with running times $2^{\mathcal{O}(\mathrm{vc} \log \mathrm{vc})} n^{\mathcal{O}(1)}$, $2^{\mathcal{O}(\mathrm{vi}^2 \log \mathrm{vi})} n^{\mathcal{O}(1)}$, and $2^{\mathcal{O}(k \log k)} n^{\mathcal{O}(1)}$. The main ingredient that makes our algorithms faster is a Turing reduction to edge-weighted $b$-Matching.