Optimizing Distances for Multi-Broadcast in Temporal Graphs

Daniele Carnevale, Gianlorenzo D'Angelo

Keywords: Temporal graphs, Approximation algorithms, Scheduling, Combinatorial optimization, Routing

TL;DR: We study scheduling edge availability in temporal graphs to optimize the worst-case temporal distance from source vertices to all others, proving tractability and hardness results, and identifying structural conditions for efficient solutions.

Abstract:

Temporal graphs model networks in which connections change over time, with edges available only at specific moments and paths following the flow of edge availabilities. We introduce the $\mathcal{D}$-Temporal Multi-Broadcast ($\mathcal{D}$-TMB) problem, which asks for scheduling the availability of edges of a given graph so that a predetermined subset of vertices, called sources, can reach all other vertices while optimizing the worst-case temporal distance $\mathcal{D}$ from any source. Each edge can be scheduled to be available for no more than a given number of time slots, called multiplicity. This problem has several applications in logistics, multi-agent information spreading, and wireless networks, where the networks are inherently dynamic.

We characterize the computational complexity and approximability of $\mathcal{D}$-TMB under six different definitions of temporal distance $\mathcal{D}$, namely Earliest-Arrival (EA), Latest-Departure (LD), Fastest-Time (FT), Shortest-Traveling (ST), Minimum-Hop (MH), and Minimum-Waiting (MW). For a single source, $\mathcal{D}$-TMB can be solved in polynomial time for EA and LD. While tractability for EA was already known, we additionally show that the LD measure is also tractable. In contrast, for the other four temporal distances, we prove that $\mathcal{D}$-TMB is NP-hard and hard to approximate within a factor that depends on the adopted distance function. We provide a matching approximation algorithm for $\mathcal{D} \in \{ \text{FT}, \text{MW} \}$. In the case of multiple sources, we show that even deciding whether a feasible solution exists is NP-complete for any $\mathcal{D}$, which implies that the problem is inapproximable within any factor, unless P = NP. The hardness holds even for two sources. We complement this negative result by identifying structural conditions that guarantee tractability for EA and LD for any number of sources: each edge has multiplicity at least equal to the number of sources, or the underlying graph is a tree and each edge has multiplicity at least two.

Area: Search, Optimization, Planning, and Scheduling (SOPS)

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Submission Number: 1044