SD-ZFS: RL for Minimum Zero-Forcing Sets

• The paper adapts S2V-DQN to model the sequential construction of a zero-forcing set as a Markov decision process with a per-step cost, addressing an NP-hard problem.
• It employs specialized structure2vec embeddings and deep Q-networks to capture global graph properties such as clustering and hub connectivity.
• The framework generalizes across diverse graph families, consistently outperforming greedy heuristics on random, scale-free, and real-world networks.

SD-ZFS, short for Structure2Vec-DQN for Zero-Forcing Sets, is a reinforcement-learning framework for the minimum zero-forcing set problem on simple undirected graphs. In this problem, an initial blue set propagates color through a network under the zero-forcing color-change rule, and the objective is to find a smallest set whose closure colors the entire graph. The framework adapts S2V-DQN to this combinatorial setting, evaluates generalization and transfer across graph families with varying structure, and is reported to be effective relative to both optimal solutions on small instances and a greedy heuristic across several synthetic and real-world datasets (Halley et al., 16 Jun 2026).

1. Problem definition and graph-theoretic setting

Let $G=(V,E)$ be a simple undirected graph with vertex set $V$ and edge set $E$. An initial set $S\subseteq V$ is colored blue, while the remaining vertices $T=V\setminus S$ are white. The zero-forcing color-change rule is then applied repeatedly: if a blue vertex $u$ has exactly one white neighbour $v$, then $u$ forces $v$ to blue. The closure of $S$ under zero-forcing is defined by

$V$0

and $V$1. A set $V$2 is a zero-forcing set if $V$3. The minimum zero-forcing number is

$V$4

The paper characterizes the task as a graph coloring problem with applications in network science, network control, and designing logical circuits. It also states that finding the minimum zero-forcing set is NP-hard. These properties motivate heuristic and learning-based methods rather than exact optimization on general graphs (Halley et al., 16 Jun 2026).

2. Markov decision process formulation

SD-ZFS models the sequential construction of a zero-forcing set as a Markov decision process. The state $V$5 consists of the graph $V$6, the partial solution $V$7, and the current set of blue vertices $V$8. Each vertex $V$9 is assigned a feature vector $E$0 indicating whether $E$1 has already been selected into $E$2 and whether $E$3 is currently blue.

An action $E$4 selects one white, unselected vertex $E$5 to add to the solution. The transition adds $E$6 to $E$7 and then performs zero-forcing propagation, producing

$E$8

The reward is a per-step cost of $E$9 for each addition,

$S\subseteq V$0

with no further penalty at termination. Episodes terminate when $S\subseteq V$1. The optimization target is the discounted return

$S\subseteq V$2

with $S\subseteq V$3 close to $S\subseteq V$4, so that shorter episodes correspond to smaller final sets.

This formulation makes the cardinality objective operational through sequential decision making: every unnecessary addition incurs an explicit cost. A plausible implication is that the method is designed to learn not merely immediate closure gains, but also action sequences whose delayed forcing consequences reduce $S\subseteq V$5 (Halley et al., 16 Jun 2026).

3. Adapted S2V-DQN architecture

The framework combines a Structure2Vec graph embedding with a Deep Q-Network. In the Structure2Vec component, each node begins with the feature vector $S\subseteq V$6, and message passing is performed for $S\subseteq V$7 mean-field iterations: $S\subseteq V$8 for $S\subseteq V$9, with $T=V\setminus S$0. Here $T=V\setminus S$1 is any edge feature. After $T=V\setminus S$2 hops, $T=V\setminus S$3 encodes local structure. A graph-level representation is then obtained by sum pooling,

$T=V\setminus S$4

The DQN component assigns a Q-score to each legal action $T=V\setminus S$5: $T=V\setminus S$6 where $T=V\setminus S$7 denotes concatenation. The policy is greedy,

$T=V\setminus S$8

Relative to generic S2V-DQN, the adaptation is specified in three ways: the node features $T=V\setminus S$9 are specialized to encode selection and blue-state information; zero-forcing propagation is enforced by the environment after each action; and redundant selections are pruned at the end of each episode. The paper’s interpretation is that the message-passing embeddings capture structural properties such as hubs and clustering and use them to guide node selection beyond purely local gain (Halley et al., 16 Jun 2026).

4. Training setup and model variants

Three models are trained on distinct graph distributions.

Model	Training distribution	Episodes
SD-ZFS ER	Erdős–Rényi $u$0	5,760
SD-ZFS BA	Barabási–Albert $u$1	6,765
SD-ZFS FB	small real-world Facebook ego-networks	12,849

The common hyperparameters are: $u$2-step returns with $u$3; latent dimension of $u$4 equal to $u$5; hidden size in the Q-MLP equal to $u$6; discount factor $u$7; $u$8-greedy exploration with $u$9 linearly decayed from $v$0 to $v$1; Adam with learning rate $v$2; replay buffer capacity $v$3; minibatch size $v$4; and target network update every $v$5 steps. No special data augmentation or curriculum was used beyond re-sampling fresh graphs each episode (Halley et al., 16 Jun 2026).

This training regime emphasizes distribution-specific learning while also testing transfer across graph families. Because fresh graphs are re-sampled each episode, the setup is explicitly oriented toward learning policies over graph distributions rather than memorizing fixed instances.

5. Evaluation metrics and reported results

Evaluation is performed on a test set $v$6 of graphs using three metrics: $v$7

$v$8

and

$v$9

On small graphs, the reported values are as follows.

Model	Small ER	Small BA
SD-ZFS ER	Dev = +11.45%, Diff = +4.47, Approx = 1.048	—
SD-ZFS BA	Dev = +10.39%, Diff = +4.06, Approx = 1.060	—
SD-ZFS FB	Dev = −7.85%, Diff = −3.13, Approx = 1.090	—

The accompanying discussion states that, on Small_ER, SD-ZFS ER beats the greedy heuristic by $u$0 nodes on average, at an approximation ratio $u$1, and that all three models closely match or outperform the greedy solution on small scale-free graphs. This suggests that the paper’s narrative interpretation should be read together with the reported Difference and Deviance sign convention (Halley et al., 16 Jun 2026).

On large and real-world graphs, the reported outcomes are:

Model	Large ER	Large BA	COLLAB	IMDB	REDDIT
SD-ZFS ER	Dev +0.71%, Diff +1.37	Dev −0.64%, Diff −2.77	Dev −0.30%, Diff −0.13	0.00%	Dev −0.95%, Diff −1.56
SD-ZFS BA	Dev +1.59%, Diff +6.23	Dev +1.18%, Diff +1.24	Dev −0.02%, Diff −0.07	0.00%	Dev −0.29%, Diff −0.67
SD-ZFS FB	Dev −3.06%, Diff −11.5	Dev −2.13%, Diff −6.41	Dev +0.45%, Diff +0.20	0.00%	Dev −0.80%, Diff −1.21

The paper highlights several specific findings. On large Erdős–Rényi graphs, SD-ZFS ER improves by $u$2 nodes with Dev $u$3. SD-ZFS BA generalizes from scale-free to random graphs and, on Large ER, wins $u$4 of the time. On COLLAB real-world networks, SD-ZFS FB slightly outperforms greedy with Dev $u$5. On IMDB-BINARY, all methods essentially reach optimal on small instances of at most $u$6 nodes. On REDDIT-BINARY, none of the models beat greedy on average; the dataset is described as dominated by hub-and-spoke components, for which $u$7 makes sophisticated selection inconsequential. The reported approximation-ratio plots and the ER_Scale and BA_Scale experiments are said to show consistent outperformance of greedy as graph size grows (Halley et al., 16 Jun 2026).

6. Structural interpretation, transfer behavior, and limitations

The central conclusion is that SD-ZFS learns problem-specific heuristics that substantially improve on the classic greedy closure-based algorithm, especially on random and scale-free graphs. The reported explanation is that message-passing embeddings capture global structure, including hubs and clustering, and thereby support node choices that are not reducible to immediate local closure effects.

Transfer across graph families is a prominent feature of the reported results. Models trained on one structure, such as Barabási–Albert graphs, can successfully transfer to others, such as Erdős–Rényi graphs, although performance degrades on extreme cases, particularly hub-and-spoke networks. This makes the method neither uniformly dominant nor structure-agnostic: its behavior depends materially on the relation between training distribution and test topology (Halley et al., 16 Jun 2026).

The paper also delineates graph classes where the room for improvement is intrinsically limited. Dense graphs, or graphs near clique-like structure, force $u$8, so all heuristics converge. Very sparse tree-like graphs, including paths and trees, are solvable in $u$9 by classical methods. A common misconception would be that a learned policy should outperform simpler procedures on every network type; the reported REDDIT-BINARY results, together with the observations on clique-like and hub-and-spoke regimes, indicate that structural degeneracy can make sophisticated selection strategies largely inconsequential.

Within those boundaries, SD-ZFS is presented as an end-to-end reinforcement-learning-based heuristic that learns from graph topology, scales to hundreds or thousands of vertices, and yields solutions that on average are closer to optimal than the best known greedy methods (Halley et al., 16 Jun 2026).