We prove that given any $α$-approximation LOCAL algorithm for Minimum Dominating Set (MDS) on planar graphs, we can construct an $f(g)$-round $(3α+1)$-approximation LOCAL algorithm for MDS on graphs embeddable in a given Euler genus-$g$ surface. Heydt et al. [European Journal of Combinatorics (2025)] gave an algorithm with $α=11+\varepsilon$, from which we derive a $(34 +\varepsilon)$-approximation algorithm for graphs of genus $g$, therefore improving upon the current state of the art of $24g+O(1)$ due to Amiri et al. [ACM Transactions on Algorithms (2019)]. It also improves the approximation ratio of $91+\varepsilon$ due to Czygrinow et al. [Theoretical Computer Science (2019)] in the particular case of orientable surfaces. We generalize this result into two directions: (1) by considering other graph problems studied in Distributed Computing such as Minimum $k$-Tuple Dominating Set, for which constant-round approximation algorithms were known for planar graphs, but not for graphs of bounded genus; and (2) by considering graph classes beyond bounded genus graphs, called locally nice, and relying on the asymptotic dimension of the class. We prove these results by a series of meta-theorems about cuttable minimization problems with constant-round approximation LOCAL algorithms. Roughly speaking, in cuttable problems, one can systematically extract small subgraphs whose solutions are in proportion to the global solution restricted to the neighbourhood of the subgraph.

翻译：我们证明，给定任何用于平面图上最小支配集（MDS）的$α$-近似LOCAL算法，可以构造一个$f(g)$轮的$(3α+1)$-近似LOCAL算法，用于可嵌入给定欧拉亏格$g$曲面上的图。Heydt等人[European Journal of Combinatorics (2025)]给出了一种$α=11+\varepsilon$的算法，由此我们推导出针对亏格$g$图的$(34 +\varepsilon)$-近似算法，从而改进了Amiri等人[ACM Transactions on Algorithms (2019)]当前最优的$24g+O(1)$结果。对于可定向曲面的特例，它也改进了Czygrinow等人[Theoretical Computer Science (2019)]的$91+\varepsilon$近似比。我们将这一结果推广到两个方向：（1）考虑分布式计算中研究的其他图问题，如最小$k$元支配集，对于平面图已知存在常数轮近似算法，但对于有界亏格图则未必；（2）考虑超越有界亏格图的图类（称为局部良图），并依赖该类图的渐近维数。我们通过一系列关于具有常数轮近似LOCAL算法的可切割最小化问题的元定理来证明这些结果。大致而言，在可切割问题中，可以系统地提取子图，其解与全局解在该子图邻域上的限制成正比。