---
title: 'FML: Fair Minimum Labeling in Temporal Networks'
url: https://www.emergentmind.com/topics/fair-minimum-labeling-fml
type: topic
---

# FML: Fair Minimum Labeling in Temporal Networks

Searching arXiv for recent papers on Fair Minimum Labeling and related uses of the acronym FML.
Fair Minimum Labeling (FML) is a temporal network design problem with explicit fairness constraints. It considers a static undirected graph \(G=(V,E)\) in which activating an edge at a specific time incurs resource cost, and a temporal labeling specifies at which discrete time steps each edge is active. The objective is to choose when to activate which edges so that each defined group of nodes has sufficient temporal access to a designated terminal or terminal set, while minimizing the total number of edge activations. In the formulation introduced in "Fair Minimum Labeling: Efficient Temporal Network Activations for Reachability and Equity" [2510.03899], FML captures the trade-off between resource efficiency and equitable access in settings such as distributed data collection, update dissemination in edge-cloud systems, fair service restoration in critical infrastructure, and fair multi-source data aggregation for training a shared model.

## 1. Formal definition and optimization objective

FML starts from a static graph \(G=(V,E)\) with undirected edges
\[
E \subseteq \{\{u,v\} \mid u,v \in V,\, u\neq v\},
\]
together with a color function
\[
c : V \rightarrow \mathcal{P}(\mathcal{C}),
\]
where \(\mathcal{C}\) is a finite set of colors and \(c(v)\subseteq \mathcal{C}\) is the set of colors assigned to node \(v\). For a single color \(c\), the node set of that color is
\[
V_c = \{v \in V \mid c \in c(v)\}.
\]

A temporal labeling is a map
\[
\lambda : E \rightarrow \mathcal{P}(\mathbb{N}),
\]
assigning to each static edge \(e\) the set \(\lambda(e)\) of timestamps at which that edge is active. This induces a temporal graph \(G_\lambda = (V,E_\lambda)\), where
\[
E_\lambda = \{ (\{u,v\},\tau) \mid \{u,v\} \in E,\ \tau \in \lambda(\{u,v\})\}.
\]
The total activation cost is the total number of temporal edges,
\[
|\lambda| = |E_\lambda| = \sum_{e\in E} |\lambda(e)|,
\]
and this is the quantity minimized in FML [2510.03899].

The formal problem definition is: given a colored static graph \(G=(V,E,c)\), a set of terminals \(\mathcal{T}\subseteq V\), an integer \(\rho\in\mathbb{N}\), and a requirement function \(f_c : V \to \mathbb{R}\) for each color \(c\in\mathcal{C}\), find a temporal labeling \(\lambda:E\to\mathcal{P}(\mathbb{N})\) minimizing
\[
|\lambda| = \sum_{e\in E}|\lambda(e)|
\]
such that for each color \(c\in\mathcal{C}\), the number of nodes in \(V_c\) that can temporally reach at least \(\rho\) terminals \(t\in\mathcal{T}\) under \(\lambda\) is at least \(f_c(V_c)\) [2510.03899].

In the simpler single-terminal, two-group variant, the input consists of \(G=(V,E)\), terminal \(t\in V\), disjoint color sets \(B,R \subseteq V\), and parameter \(\alpha\in[0,1]\). The optimization becomes
\[
\min_{\lambda} |\lambda| = \sum_{e\in E}|\lambda(e)|,
\]
subject to
\[
\left|\{b\in B : t\text{ is temporally reachable from }b \text{ in }G_\lambda\}\right| \ge \alpha|B|,
\]
and
\[
\left|\{r\in R : t\text{ is temporally reachable from }r \text{ in }G_\lambda\}\right| \ge \alpha|R|.
\]
This special case is the main algorithmic setting developed in the paper [2510.03899].

## 2. Temporal reachability and the fairness model

Temporal reachability is defined through time-respecting paths. A temporal path from \(u\) to \(v\) is a sequence
\[
(\{v_0,v_1\},\tau_1), (\{v_1,v_2\},\tau_2),\dots,(\{v_{p-1},v_p\},\tau_p)
\]
with \(v_0=u\), \(v_p=v\), and strictly increasing timestamps \(\tau_1 < \dots < \tau_p\). Node \(v\) is temporally reachable from \(u\) if such a path exists. The reachability indicator is
\[
r(u,v) =
\begin{cases}
1 & \text{if } v \text{ is temporally reachable from } u,\\
0 & \text{otherwise,}
\end{cases}
\]
with \(r(v,v)=1\) by convention. For sets \(U,W\subseteq V\), aggregate reachability is
\[
r(U,W) = \sum_{u\in U}\sum_{v\in W} r(u,v).
\]
These definitions make FML explicitly path-based and temporal, rather than purely static [2510.03899].

The fairness model is group-level and coverage-based. Groups are the color classes \(\{V_c\}_{c\in\mathcal{C}}\); in the general definition, nodes can belong to multiple groups, while the algorithmic treatment focuses on disjoint groups \(B\) and \(R\) plus uncolored nodes. Uncolored nodes may serve as intermediates but have no coverage requirement. For a color \(c\), access is measured by the set
\[
S_c(\lambda) = \{v\in V_c : \exists \text{ a temporal path from } v \text{ to some terminal } t\in\mathcal{T} \text{ in } G_\lambda\},
\]
and the requirement is
\[
|S_c(\lambda)| \ge f_c(V_c).
\]

In the main two-group setting, the requirement functions are proportional:
\[
f_B(B) = \alpha |B|,\qquad f_R(R) = \alpha |R|,
\]
so at least an \(\alpha\)-fraction of each group must be able to reach the terminal. The paper emphasizes that one is not allowed to satisfy coverage solely from one group; both must be adequately served [2510.03899]. This distinguishes FML from formulations that optimize only total coverage.

The paper also outlines a bicriteria fairness notion arising from the scalable approximation scheme. In that variant, coverage may be violated by a multiplicative factor
\[
\xi = (1+\varepsilon)^{H+1},
\]
where \(H\) is the height of an embedding tree. The guarantee becomes
\[
\text{covered}_B \ge \frac{\alpha|B|}{\xi},\qquad
\text{covered}_R \ge \frac{\alpha|R|}{\xi}.
\]
This is still group-level and deterministic in the resulting temporal graph, but it relaxes the exact quota by a controlled multiplicative factor [2510.03899].

## 3. Complexity, hardness, and approximation algorithms

FML is computationally hard. The paper states that FML generalizes both Minimum Labeling (ML) and Minimum Steiner Labeling (MSL), and from this establishes NP-hardness in general. More sharply, the main complexity theorem states that the FML problem with a single terminal is NP-complete, and that no \(((1-\epsilon)\log n)\)-approximation algorithm exists unless
\[
NP \subseteq DTIME[n^{\mathcal{O}(\log \log n)}].
\]
The inapproximability matches classical Set Cover hardness [2510.03899].

The reduction uses a single color on all element nodes, a single terminal \(r\), set nodes connected to \(r\) through chains of length \(L\), and element nodes attached to the set nodes corresponding to containing sets. The requirement is that all colored nodes reach \(r\). The construction makes the cost of choosing a set node proportional to labeling an entire chain, so feasible low-cost FML solutions correspond to small set covers. A plausible implication is that FML should be viewed not merely as a temporal connectivity problem, but as a temporally constrained fair covering problem with explicit combinatorial hardness inherited from Set Cover [2510.03899].

The positive algorithmic results target the single-terminal, two-group case. The main framework uses a probabilistic tree embedding, specifically FRT, to embed the graph metric into a random tree metric with dominating distances and \(O(\log n)\) expected stretch. The algorithm then roots the sampled tree at the terminal \(t\), solves the FML instance on the tree, and projects the resulting temporal labeling back to the original graph [2510.03899].

On trees, the exact dynamic program maintains labels of the form
\[
(b,r,c),
\]
where \(b\) is the number of blue nodes in the subtree temporally connected to the current node, \(r\) is the analogous red count, and \(c\) is the total weighted cost of labeled edges in the subtree. At each node, for each possible pair \((b,r)\), only the label with minimum cost \(c\) is kept. The resulting exact algorithm solves tree FML in time
\[
O(n^5).
\]
The faster bicriteria variant geometrically buckets the count dimensions, reducing the number of labels per node to
\[
O\left( \frac{1}{\varepsilon^2}\log^2 n \right),
\]
and obtains a weighted-tree solution whose cost is no larger than optimal while degrading coverage by at most a factor \((1+\varepsilon)^{H+1}\) along a height-\(H\) tree [2510.03899].

These tree solutions yield the main graph-level guarantees. Using exact tree DP gives exact fairness and expected activation cost
\[
\mathbb{E}[|\lambda_G|] \le O(\log n)\,k_G^*,
\]
where \(k_G^*\) is the optimal FML cost in the original graph. Using the bucketing DP gives a randomized bicriteria \((O(\log n),\xi)\) approximation with the same expected cost bound and coverage at least \((\alpha|B|)/\xi\) and \((\alpha|R|)/\xi\), with expected running time
\[
O\bigl(n^2 + n\varepsilon^{-4}\log^4 n\bigr).
\]
Since the paper also proves \(\Omega(\log n)\)-hardness of approximation, these \(O(\log n)\) guarantees are best possible in the worst case under the stated complexity assumptions [2510.03899].

## 4. Relations to adjacent optimization problems

FML is explicitly positioned relative to several established problems. It generalizes Minimum Labeling (ML) and Minimum Steiner Labeling (MSL). In the reduction described in the appendix, ML is recovered by taking a single color for all nodes, terminal set \(\mathcal{T}=V\), and requirement \(f_{c_1}(V)=|V|^2\), thereby requiring temporal connectivity for all ordered pairs. MSL is recovered by coloring only the terminals and setting the requirement to \(|\mathcal{T}|^2\), so that minimizing labels corresponds to temporal connection of the specified terminal set [2510.03899].

The paper also draws a conceptual connection to Set Cover and fair variants of Set Cover. In this analogy, activating specific temporal paths plays the role of selecting sets, and group-specific reachability quotas play the role of fair coverage constraints. It likewise relates FML to temporal Steiner tree and temporally connected subgraph formulations, but distinguishes FML by the fact that only a subset of group nodes must connect to terminals, different groups have separate quotas, and uncolored nodes may function as Steiner nodes [2510.03899].

A recurrent source of terminological ambiguity is the acronym “FML.” In "Measure Contribution of Participants in Federated Learning" [1909.08525], FML denotes Federated Machine Learning rather than Fair Minimum Labeling. That paper studies contribution measurement and fair credit allocation among multiple parties, using deletion-based grouped instance influence for Horizontal FML and Shapley group values for Vertical FML [1909.08525]. The overlap is therefore lexical, not definitional. A plausible implication is that references to “FML” in the literature require disambiguation between a temporal-network optimization problem and a federated-learning collaboration setting.

A second potential confusion arises with "FAIRLABEL: Correcting Bias in Labels" [2311.00638]. The supplied material explicitly states that it does not contain any definition of FAIRLABEL, any mention of “Fair Minimum Labeling (FML),” or any problem formulation with \((X,A,Y)\), disparate impact, or fairness constraints. It also states that the term “FML” is not defined or used anywhere in the provided document [2311.00638]. This supports an objective clarification: Fair Minimum Labeling is not a label-correction method in the supplied FAIRLABEL material, and no explicit mapping between the two is supported there.

## 5. Experimental evaluation and learning-integrated use

The principal empirical demonstration of FML is a fair multi-source data aggregation task for training a shared model. The graph is a random geometric graph with \(|V|=1024\) and radius \(0.2\), with one node designated as the terminal \(t\). Nodes in \(B\) and \(R\) are data sources belonging to two groups assigned spatially near versus far from \(t\). Each source generates labeled points \((\mathbf{x},y)\in\mathbb{R}^2\times\{0,1\}\). For group \(B\),
\[
\mathbf{x}\sim \mathcal{N}(\mu,\sigma^2 I),\quad \sigma=0.55,\quad \mu\in\{(-2,1),(2,1)\},
\]
with label \(y=\mathbf{1}\{x_1>0\}\). For group \(R\),
\[
\mathbf{x}\sim \mathcal{N}(\mu,\sigma^2 I),\quad \sigma=0.55,\quad \mu\in\{(1,-2),(1,2)\},
\]
with label \(y=\mathbf{1}\{x_2<0\}\). The paper notes that each group therefore has an orthogonal classification rule, so training primarily on one group leads to poor performance on the other [2510.03899].

Once an activation plan is chosen, each group node that can temporally reach \(t\) contributes its data to training. The learning model is a small MLP with input dimension \(2\), two hidden layers with \(32\) ReLU units each, and a single sigmoid output. Training uses Adam with learning rate \(0.01\), batch size \(64\), weight decay \(10^{-2}\), and \(100\) epochs with early stopping. Evaluation reports \(\text{Acc}_B\), \(\text{Acc}_R\), \(\text{coverage}_B\), \(\text{coverage}_R\), and cost, averaged over \(10\) runs [2510.03899].

The algorithms compared are FMLapprox, which uses FRT embedding with the exact tree DP, and FMLbiApprox, which uses the bucketing DP. Baselines are Greedy, Closest, and Alternating. Greedy is fairness-agnostic and iteratively activates shortest paths from the terminal to the closest colored node until total coverage over \(B\cup R\) reaches \(\alpha(|B|+|R|)\). Closest is fair but naive, greedily activating shortest paths to the nearest uncovered colored node until both groups individually reach \(\alpha\) coverage. Alternating alternates between covering the closest uncovered blue node and the closest uncovered red node [2510.03899].

The reported results show a pronounced fairness–cost trade-off. Greedy attains cost \(88.3\pm 6.25\) but coverage \(B:64.0\pm0.0\), \(R:0.0\pm0.0\), with \(\text{Acc}_B = 1.00\pm0.00\) and \(\text{Acc}_R = 0.27\pm0.17\). Closest attains cost \(224.0\pm 0\), coverage \(B:\approx 44.3\), \(R:32.0\), and accuracies \(0.97\pm0.04\) and \(0.94\pm0.08\). Alternating attains cost \(212.8\pm1.81\), coverage \(B:\approx 34.2\), \(R:32.0\), and accuracies approximately \(0.96\) and \(0.94\). FMLapprox attains cost \(74.3\pm4.62\), coverage \(B:33.5\pm0.53\), \(R:32.3\pm0.48\), accuracies \(0.98\pm0.03\) and \(0.93\pm0.08\), and runtime approximately \(69.2\) s. FMLbiApprox attains cost \(73.9\pm3.48\), coverage \(B:33.1\pm0.99\), \(R:31.3\pm0.48\), accuracies \(0.97\pm0.03\) and \(0.94\pm0.08\), and runtime approximately \(8.5\) s [2510.03899].

These experiments support two points stated in the paper: fairness-agnostic minimization can produce severe under-coverage of one group, and explicitly solving the temporal activation problem can achieve almost perfectly balanced coverage at substantially lower cost than fair heuristics. The paper also reports scaling experiments on Barabási–Albert graphs with \(n\in\{256,512,1024,2048,4096\}\), showing that FMLbiApprox consistently finishes and is up to an order of magnitude faster than the exact method, and a Pokec experiment on a \(20{,}000\)-node induced subgraph with \(201{,}900\) edges, where FMLbiApprox attains runtime \(\approx 3625 \pm 421\) seconds with \(\varepsilon=0.01\) and achieved coverage accuracy \(93.1\%\), and runtime \(\approx 4662 \pm 991\) seconds with \(\varepsilon=0.001\) and achieved coverage accuracy \(99.3\%\) [2510.03899].

## 6. Scope, limitations, and research directions

The paper explicitly notes several limitations. First, the strongest approximation and dynamic-programming results are developed for the case of one terminal and two groups; extending them to multiple terminals, more groups, or richer fairness notions is left for future work. It does sketch that, for more than two groups, DP labels generalize to \((g_1,\dots,g_k,c)\), with state space growing as \(O(n^k)\) and runtime as \(O(n^{2k+1})\), which indicates a rapid combinatorial blow-up [2510.03899].

Second, the exact tree DP has high polynomial complexity, and only the bicriteria approximation scales to tens of thousands of nodes. The paper suggests that further optimization may come from exploiting structural properties such as small treewidth or bounded tree height, and from more aggressive sparsification [2510.03899]. Third, the framework assumes centralized control over all edge activations. The paper states that this may not hold in decentralized systems or strict privacy settings, making distributed or online analogues of FML open problems [2510.03899].

Fourth, groups and coverage thresholds are exogenous inputs rather than learned or adaptively updated quantities. The authors identify extensions to data-driven group definitions or adaptive fairness as open directions [2510.03899]. A plausible implication is that the current formulation is best suited to settings where protected groups, service classes, or source categories are operationally specified in advance.

More broadly, the relation to federated-learning contribution measurement illustrates that FML occupies a distinct but complementary position in learning systems. The federated-learning paper uses utility attribution to allocate fair credit after collaborative model training [1909.08525], whereas Fair Minimum Labeling designs the underlying temporal communication schedule so that each group has sufficient access before or during learning [2510.03899]. This suggests a layered perspective in which temporal access fairness and post hoc contribution fairness address different stages of a shared learning pipeline.

Source: https://www.emergentmind.com/topics/fair-minimum-labeling-fml