Approximating Optimal Labelings for Temporal Connectivity

TL;DR

This work studies the Minimum Aged Labeling (MAL) problem on temporal graphs, where edge availability is scheduled by time labels under a lifetime constraint $a$, with the objective of temporal connectivity using as few labels as possible. It establishes strong hardness results, showing no polynomial-time $O(\log n)$-approximation for fixed $a\ge 2$ and no $2^{\log^{1-\varepsilon} n}$-approximation for fixed $a\ge 3$ unless unlikely complexity collapses occur, and shows these bounds extend to the related DCSS problem. On the algorithmic side, it provides approximation strategies that depend on the relation between $a$ and graph diameter $D_G$ (and radius $R_G$), including sublinear guarantees such as $O(\sqrt{n\log n})$ for $a\ge\lceil \tfrac{3}{2}D_G\rceil$ and $O((D_G n \log^2 n)^{1/3})$ for $a\ge\lceil \tfrac{5}{3}D_G\rceil$, along with several DCSS-based approaches and regime-specific results. The paper also connects MAL to DCSS, transfers hardness and approximation bounds between them, and extends the discussion to directed MAL (DMAL) and Steiner-type variants (MSL, MASL). Overall, it maps the complexity landscape of MAL and offers practical approximation schemes across multiple parameter regimes with implications for logistics, scheduling, and information diffusion in temporal networks.

Abstract

In a temporal graph the edge set dynamically changes over time according to a set of time-labels associated with each edge that indicates at which time-steps the edge is available. Two vertices are connected if there is a path connecting them in which the edges are traversed in increasing order of their labels. We study the problem of scheduling the availability time of the edges of a temporal graph in such a way that all pairs of vertices are connected within a given maximum allowed time $a$ and the overall number of labels is minimized. The problem, known as \emph{Minimum Aged Labeling} (MAL), has several applications in logistics, distribution scheduling, and information spreading in social networks, where carefully choosing the time-labels can significantly reduce infrastructure costs, fuel consumption, or greenhouse gases. The problem MAL has previously been proved to be NP-complete on undirected graphs and \APX-hard on directed graphs. In this paper, we extend our knowledge on the complexity and approximability of MAL in several directions. We first show that the problem cannot be approximated within a factor better than $O(\log n)$ when $a\geq 2$, unless $\text{P} = \text{NP}$, and a factor better than $2^{\log ^{1-ε} n}$ when $a\geq 3$, unless $\text{NP}\subseteq \text{DTIME}(2^{\text{polylog}(n)})$, where $n$ is the number of vertices in the graph. Then we give a set of approximation algorithms that, under some conditions, almost match these lower bounds. In particular, we show that the approximation depends on a relation between $a$ and the diameter of the input graph. We further establish a connection with a foundational optimization problem on static graphs called \emph{Diameter Constrained Spanning Subgraph} (DCSS) and show that our hardness results also apply to DCSS.

Figures (6)

• Figure 1: The scheduled time of trips are reported close to the edges. 1st schedule: All the 6 scheduled trips are needed to deliver all parcels and the last parcel is delivered at 9am. 2nd schedule: All the parcels are deposited in $W$ with the 3 trips at 6am and then are delivered from $W$ to the right cities at 7am; still 6 trips are needed but the last parcel is delivered at 7am. 3rd schedule: The parcels leaving cities $A$ and $C$ are deposited in $W$ with the trips at 6am; the single trip to $B$ at 7am brings the parcels directed to $B$ previously deposited at 6am and deposits the parcels from $B$ in $W$; finally, the trips at 8am deliver the parcels directed to $A$ and $C$. Only 5 trips are needed, but the last parcel is delivered at 8am.
• Figure 2: The graph $G$ constructed starting from a SC instance $(U,\mathcal{C})$ in the proof of Theorem \ref{['thm:malinapprox']}.
• Figure 3: The labeling $\lambda$ constructed from an optimal set cover in the proof of Lemma \ref{['lem:mal2']}. Each non-dashed edge in the MAL feasible solution is assigned the labels $\{1,2\}$.
• Figure 4: Example of the construction used in Theorem \ref{['thm:dcssinapprox']} for $d = 7$.
• Figure 5: The graph used in the proof of Theorem \ref{['thm:stronginapprox']}. Edges between a vertex and a set indicate that the vertex is connected to all vertices in the set.

Theorems & Definitions (53)

• Definition 1
• Definition 2
• Lemma 1
• proof
• Theorem 1
• proof
• Theorem 2
• Theorem 3: DinurSteurer, Corollary 1.5
• Lemma 2
• proof