Optimizing Distances for Multi-Broadcast in Temporal Graphs

TL;DR

The paper addresses the $D$-Temporal Multi-Broadcast ($D$-TMB) problem on temporal graphs, where edge availability is scheduled under multiplicity constraints to minimize the worst-case $D$-distance from a source set to all vertices, across six distance definitions $D \in \{EA, LD, FT, ST, MH, MW\}$. It proves a fundamental equivalence between $D$-TMB and $D$-ReachFast, enabling a unified analysis of edge-shifting and edge-labeling paradigms, and provides a complete complexity and approximability landscape: single-source instances are poly-time for $EA$ and $LD$ but NP-hard (with strong inapproximability) for $FT$, $ST$, $MH$, and $MW$, with concrete approximation results for $FT$ and $MW$. For multiple sources, feasibility becomes NP-hard to decide even with two sources and a single label per edge, but tractability is recovered under structural conditions that ensure poly-time solvability for $EA$ and $LD$ (e.g., multiplicity $\ge |S|$ or tree topology with $\mu(e) \ge 2$ on relevant edges). The results illuminate the limits of scheduling vs labeling in temporal networks and identify practical graph classes (notably trees) and multiplicity regimes that permit efficient optimization, with implications for logistics and information dissemination in dynamic networks.

Abstract

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.

Figures (6)

• Figure 1: Instance of $\mathop{\mathrm{\mathcal{D}\xspace}}\nolimits$-TMB modeling a road network. Vertices $E$ and $M$ are sources (suppliers); each label $(t,w)$ on an edge indicates an available connection at time $t$ with traversal time $w$. In this example we assume that at most one label can be selected per edge.
• Figure 2: Three solutions for the $\mathop{\mathrm{\mathcal{D}\xspace}}\nolimits$-Temporal Multi-Broadcast instance of Figure \ref{['fig:exampleone']}, where the aim is to minimize the maximum earliest arrival time, the maximum duration of travels, and the maximum travel time, respectively.
• Figure 3: Illustration of reduction for $\mathop{\mathrm{\mathcal{D}\xspace}}\nolimits=\mathop{\mathrm{FT}}\nolimits$ in Theorem \ref{['thm:sshard']}. The number $x$ near an edge $e$ indicates that $\mathop{\mathrm{tr}}\nolimits(e,x)=1$.
• Figure 4: Illustration of the necessary modifications in the reduction for $\mathop{\mathrm{\mathcal{D}\xspace}}\nolimits \in \{\mathop{\mathrm{ST}}\nolimits, \mathop{\mathrm{MH}}\nolimits, \mathop{\mathrm{MW}}\nolimits\}$ (from left to right, respectively) in Theorem \ref{['thm:sshard']}. Here, $a$ denotes an integer satisfying $a \geq 2,3,1$, respectively, and $b\geq 2$.
• Figure 5: Illustration of the reduction used in the proof of Theorem \ref{['thm:noapx']}. Gray vertices and edges correspond to those introduced by subdivisions.

Theorems & Definitions (32)

• Definition 1: $\mathop{\mathrm{\mathcal{D}\xspace}}\nolimits$-ReachFast problem
• Definition 2: $\mathop{\mathrm{LD}}\nolimits$-ReachFast problem
• Definition 3: $\mathop{\mathrm{\mathcal{D}\xspace}}\nolimits$-Temporal Multi-Broadcast problem
• Definition 4: $\mathop{\mathrm{LD}}\nolimits$-Temporal Multi-Broadcast problem
• Theorem 1
• proof
• Theorem 2
• Theorem 3
• proof
• Corollary 1