Temporal graphs represent networks in which connections change over time, with edges available only at specific moments. Motivated by applications in logistics, multi-agent information spreading, and wireless networks, we introduce the D-Temporal Multi-Broadcast (D-TMB) problem, which asks for scheduling the availability of edges so that a predetermined subset of sources reach all other vertices while optimizing the worst-case temporal distance D from any source. We show that D-TMB generalizes ReachFast (arXiv:2112.08797). We then characterize the computational complexity and approximability of D-TMB under six definitions of temporal distance D, namely Earliest-Arrival (EA), Latest-Departure (LD), Fastest-Time (FT), Shortest-Traveling (ST), Minimum-Hop (MH), and Minimum-Waiting (MW). For a single source, we show that D-TMB can be solved in polynomial time for EA and LD, while for the other temporal distances it is NP-hard and hard to approximate within a factor that depends on the adopted distance function. We give approximation algorithms for FT and MW. For multiple sources, if feasibility is not assumed a priori, the problem is inapproximable within any factor unless P = NP, even with just two sources. We complement this negative result by identifying structural conditions that guarantee tractability for EA and LD for any number of sources.

翻译：时态图表示连接随时间变化的网络，其中边仅在特定时刻可用。受物流、多智能体信息传播和无线网络等应用的启发，我们提出了D-时态多源广播问题，该问题要求调度边的可用性，使得预定源点子集能够到达所有其他顶点，同时优化任意源点的最坏情况时态距离D。我们证明D-TMB推广了ReachFast问题。随后，我们基于六种时态距离定义——最早到达、最晚出发、最快时间、最短行程、最小跳数和最小等待——系统刻画了D-TMB问题的计算复杂度与近似性。针对单源情形，我们证明D-TMB在EA和LD距离下可在多项式时间求解，而对于其他时态距离度量，该问题具有NP难解性，且近似难度取决于所采用的距离函数。我们为FT和MW距离设计了近似算法。对于多源情形，若未预先假设可行性，即使仅有两个源点，该问题也不存在任何近似因子内的可近似解。我们进一步通过识别保证任意数量源点在EA和LD距离下可解的结构性条件，对这一否定结论进行了补充。