New meta-theorems turn planar graph algorithms into bounded-genus solutions

A team of six computer scientists from France, Finland, and Germany has proven a set of meta-theorems that dramatically simplify the design of distributed approximation algorithms for graphs embedded on surfaces. Their key result: given any α-approximation LOCAL algorithm for Minimum Dominating Set (MDS) on planar graphs, one can automatically construct an f(g)-round (3α+1)-approximation algorithm for MDS on graphs embeddable on a fixed Euler genus-g surface. Plugging in the state-of-the-art planar algorithm by Heydt et al. (α=11+ε), the authors obtain a (34+ε)-approximation for all genus g — improving over the prior best 24g+O(1) from Amiri et al. (2019) and the 91+ε bound for orientable surfaces from Czygrinow et al. (2019).

The paper extends the meta-approach in two directions. First, they apply the same cuttable-problem framework to Minimum k-Tuple Dominating Set, a generalization where each vertex must be dominated by at least k distinct vertices. Previously, constant-round algorithms existed only for planar graphs; the new result yields constant-round (3α+1)-approximation for any fixed genus g. Second, they move beyond bounded genus to “locally nice” graph classes defined by certain separation properties, leveraging the asymptotic dimension of the class. Roughly speaking, a problem is “cuttable” if one can extract small subgraphs whose optimal solution size is proportional to the global solution restricted to that subgraph’s neighborhood. The authors prove meta-theorems showing that any constant-round approximate algorithm for such problems on planar graphs lifts automatically to bounded-genus surfaces and even to locally nice classes.

This work is purely theoretical but has practical implications for distributed computing (LOCAL model) where each node has only local knowledge and communication is synchronous. The ability to reuse planar algorithms without redesigning for each surface class simplifies system design in wireless sensor networks, routing, and fault-tolerance. The paper appears as arXiv:2605.19157 and spans 31 pages with 6 figures. The authors include Marthe Bonamy, Avinandan Das, Cyril Gavoille, Timothé Picavet, Jukka Suomela, and Alexandra Wesolek.