Optimizing Distances for Multi-Broadcast in Temporal Graphs

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.

💡 Research Summary

The paper introduces the D‑Temporal Multi‑Broadcast (D‑TMB) problem, a scheduling task on temporal graphs where the goal is to assign time labels to edges (subject to multiplicity constraints) so that a given set of source vertices can reach every other vertex while minimizing the worst‑case temporal distance from any source. Six widely studied temporal distance measures are considered: Earliest‑Arrival (EA), Latest‑Departure (LD), Fastest‑Time (FT), Shortest‑Traveling (ST), Minimum‑Hop (MH), and Minimum‑Waiting (MW). The authors first show that D‑TMB generalizes the ReachFast problem and is equivalent to a D‑ReachFast formulation for any distance measure, allowing them to focus on D‑TMB.

Single‑source results. For a single source (|S| = 1) the problem is polynomial‑time solvable for EA (already known) and for LD (new algorithm). For the remaining four distances (FT, ST, MH, MW) the decision version (“does a schedule achieving value ≤ K exist?”) is NP‑complete. Moreover, they prove strong inapproximability: ST and MH cannot be approximated within any factor better than 2 (APX‑hard), while FT and MW admit no polynomial‑time approximation better than an exponential function of the number of vertices or of the input weights, unless P = NP. Despite these hardness results, the authors present simple approximation algorithms for FT and MW that match the respective lower bounds, achieving the best possible approximation ratio under the proven limits.

Multiple‑source results. When two or more sources are present, the situation becomes dramatically harder. Even deciding feasibility (whether any schedule respecting multiplicities exists) is NP‑complete for every distance measure. Consequently, if feasibility is not guaranteed a priori, D‑TMB cannot be approximated within any finite factor for any of the six distances, a result that already holds for just two sources.

Tractable structural cases. The paper then identifies conditions under which the problem regains polynomial‑time solvability for EA and LD despite multiple sources. If the multiplicity of each edge is at least the number of sources (|S|), a greedy construction yields an optimal schedule. Alternatively, when the underlying static graph is a tree and every edge has multiplicity at least 2, a dynamic programming approach solves EA and LD efficiently. These structural guarantees provide practical guidance for networks that are tree‑like or have abundant edge usage capacity.

Methodology and proofs. Hardness proofs rely on reductions from classic NP‑hard problems (3‑SAT, Set Cover, Hamiltonian Path) to temporal‑graph scheduling instances, carefully encoding the chosen distance measure into the timing constraints. The exponential inapproximability for FT and MW is obtained via gap‑reductions that amplify the difference between feasible and infeasible instances using edge weights. The polynomial algorithms for EA and LD exploit the monotonicity of earliest arrival and latest departure times, constructing earliest‑arrival trees or latest‑departure schedules directly from the underlying graph.

Related work and contributions. The authors situate their work among prior studies on temporal graph manipulation (delaying, shifting), temporal branchings, and temporal spanning subgraphs. While earlier papers focused on single‑source optimization or on minimizing the sum of distances, this work uniquely studies the minimax objective across multiple sources and across a comprehensive set of distance definitions. It also extends the ReachFast framework to a multi‑source setting and provides a complete complexity landscape (polynomial, NP‑complete, APX‑hard, inapproximable) for each distance measure.

Practical implications. In logistics, the results suggest that if a company cares about the latest delivery time (EA) or wants to start shipments as late as possible without delaying overall completion (LD), efficient schedules can be computed even with many depots. However, optimizing total travel time (FT) or minimizing waiting (MW) is computationally prohibitive unless the network has a tree topology or edges can be reused many times. Similar conclusions apply to multi‑agent information spreading and energy‑constrained wireless sensor networks.

Conclusion and future directions. The paper delivers a thorough classification of D‑TMB’s computational difficulty, highlighting how the choice of temporal distance dramatically influences tractability. Open avenues include parameterized algorithms (e.g., FPT with respect to treewidth or number of sources), online versions where edge times are revealed incrementally, and empirical evaluation of the proposed approximation algorithms on real‑world temporal network data.