We consider minimum time multicasting problems in directed and undirected graphs: given a root node and a subset of $t$ terminal nodes, multicasting seeks to find the minimum number of rounds within which all terminals can be informed with a message originating at the root. In each round, the telephone model we study allows the information to move via a matching from the informed nodes to the uninformed nodes. Since minimum time multicasting in digraphs is poorly understood compared to the undirected variant, we study an intermediate problem in undirected graphs that specifies a target $k < t$, and requires the only $k$ of the terminals be informed in the minimum number of rounds. For this problem, we improve implications of prior results and obtain an $\tilde{O}(t^{1/3})$ multiplicative approximation. For the directed version, we obtain an {\em additive} $\tilde{O}(k^{1/2})$ approximation algorithm (with a poly-logarithmic multiplicative factor). Our algorithms are based on reductions to the related problems of finding $k$-trees of minimum poise (sum of maximum degree and diameter) and applying a combination of greedy network decomposition techniques and set covering under partition matroid constraints.

翻译：我们研究有向图和无向图中的最小时间多播问题：给定一个根节点和包含$t$个终端节点的子集，多播问题旨在找到所有终端节点能够接收到根节点发起消息的最小轮数。在每一轮中，我们所研究的电话模型允许信息通过已通知节点与未通知节点之间的匹配进行传播。由于有向图中的最小时间多播问题相较于无向图变体而言研究尚不充分，我们研究无向图中的一个中间问题，该问题设定目标$k < t$，并要求仅需在最小轮数内通知$k$个终端节点。针对该问题，我们改进了先前结果的推论，并得到了$\tilde{O}(t^{1/3})$的乘法近似解。对于有向图版本，我们获得了一个具有$\tilde{O}(k^{1/2})$加法近似的算法（同时包含多对数乘法因子）。我们的算法基于对相关问题的归约：寻找最小镇定度（最大度与直径之和）的$k$-树，并应用了贪心网络分解技术与划分拟阵约束下的集合覆盖方法的组合。