Fair Minimum Labeling: Efficient Temporal Network Activations for Reachability and Equity

TL;DR

The paper defines Fair Minimum Labeling (FML), a problem that minimizes temporal edge activations while guaranteeing group-wise reachability to a target set under fairness constraints. It proves NP-hardness and $\,\Omega\left(\log |V|\right)$-hardness to approximate, and develops a probabilistic tree-embedding framework (FRT) that yields an $\mathcal{O}(\log |V|)$-factor approximate solution for the two-color, single-terminal case, with a faster bicriteria variant that accepts bounded fairness violation. The approach includes an exact dynamic-programming algorithm for FML on weighted trees, a bucket-based approximation to scale up, and a principled projection back to the original graph. Empirical results on multi-source data collection tasks show that FML methods achieve near-ideal group fairness with substantially lower activation costs than baselines, demonstrating practical efficiency and scalability for resource-constrained, learning-integrated networks. The work lays the groundwork for extending to multiple terminals, more groups, weighted/online settings, and distributed deployments in real-world applications like environmental sensing, edge-cloud updates, and infrastructure restoration.

Abstract

Balancing resource efficiency and fairness is critical in networked systems that support modern learning applications. We introduce the Fair Minimum Labeling (FML) problem: the task of designing a minimum-cost temporal edge activation plan that ensures each group of nodes in a network has sufficient access to a designated target set, according to specified coverage requirements. FML captures key trade-offs in systems where edge activations incur resource costs and equitable access is essential, such as distributed data collection, update dissemination in edge-cloud systems, and fair service restoration in critical infrastructure. We show that FML is NP-hard and $Ω(\log |V|)$-hard to approximate, where $V$ is the set of nodes, and we present probabilistic approximation algorithms that match this bound, achieving the best possible guarantee for the activation cost. We demonstrate the practical utility of FML in a fair multi-source data aggregation task for training a shared model. Empirical results show that FML enforces group-level fairness with substantially lower activation cost than baseline heuristics, underscoring its potential for building resource-efficient, equitable temporal reachability in learning-integrated networks.

Figures (2)

• Figure 1: A simple toy example showing a non-equitable and an equitable labeling of a small colored graph. The labelings $\lambda_1$ and $\lambda_2$ assign time stamps $\tau_1<\tau_2<\tau_3$ to edges. Both connect 50% of the colored nodes, but only $\lambda_2$ connects both colors resulting in higher costs of three. Therefore, under a color requirement of $50\%$ for each color, only $\lambda_2$ is a valid solution for the FML problem.
• Figure 2: Comparison of FMLbiApprox and FMLapprox. The results show the mean and standard deviation as error bars over ten independent runs.

Theorems & Definitions (13)

• Definition 1: Fair Minimum Labeling (FML)
• Theorem 1
• Definition 2: Probabilistic Tree Embedding
• Theorem 2
• Theorem 3
• Theorem 4
• proof : Proof of \ref{['lemma:generalization']}
• proof : Proof of \ref{['theorem:maincomplexity']}
• proof : Proof of \ref{['theorem:runningtimetree']}
• Lemma 1