A dominating set of a graph $\mathcal{G=(V, E)}$ is a subset of vertices $S\subseteq\mathcal{V}$ such that every vertex $v\in \mathcal{V} \setminus S$ outside the dominating set is adjacent to a vertex $u\in S$ within the set. The minimum dominating set problem seeks to find a dominating set of minimum cardinality and is a well-established NP-hard combinatorial optimization problem. We propose a novel learning-based heuristic approach to compute solutions for the minimum dominating set problem using graph convolutional networks. We conduct an extensive experimental evaluation of the proposed method on a combination of randomly generated graphs and real-world graph datasets. Our results indicate that the proposed learning-based approach can outperform a classical greedy approximation algorithm. Furthermore, we demonstrate the generalization capability of the graph convolutional network across datasets and its ability to scale to graphs of higher order than those on which it was trained. Finally, we utilize the proposed learning-based heuristic in an iterative greedy algorithm, achieving state-of-the-art performance in the computation of dominating sets.

翻译：图$\mathcal{G=(V, E)}$的支配集是顶点子集$S\subseteq\mathcal{V}$，使得支配集外每个顶点$v\in \mathcal{V} \setminus S$都与该集内某个顶点$u\in S$相邻。最小支配集问题旨在寻找基数最小的支配集，这是一个经典的NP-难组合优化问题。我们提出了一种新颖的基于学习的启发式方法，利用图卷积网络计算最小支配集问题的解。我们在随机生成图与真实世界图数据集的组合上对所提方法进行了广泛实验评估。结果表明，该基于学习的方法能超越经典的贪心近似算法。此外，我们验证了图卷积网络跨数据集的泛化能力及其在高于训练图阶数的图上进行扩展的能力。最后，我们将此学习型启发式算法应用于迭代贪心算法，在支配集计算中取得了当前最优性能。