Distributed Domination on Graph Classes of Bounded Expansion

We provide a new constant factor approximation algorithm for the (connected) distance-$r$ dominating set problem on graph classes of bounded expansion. Classes of bounded expansion include many familiar classes of sparse graphs such as planar graphs and graphs with excluded (topological) minors, and notably, these classes form the most general subgraph closed classes of graphs for which a sequential constant factor approximation algorithm for the distance-$r$ dominating set problem is currently known. Our algorithm can be implemented in the \congestbc model of distributed computing and uses $\mathcal{O}(r^2 \log n)$ communication rounds. Our techniques, which may be of independent interest, are based on a distributed computation of sparse neighborhood covers of small radius on bounded expansion classes. We show how to compute an $r$-neighborhood cover of radius~$2r$ and overlap $f(r)$ on every class of bounded expansion in $\mathcal{O}(r^2 \log n)$ communication rounds for some function~$f$.% in the $\mathcal{CONGEST}_{\mathrm{BC}}$ model. Finally, we show how to use the greater power of the $\mathcal{LOCAL}$ model to turn any distance-$r$ dominating set into a constantly larger connected distance-$r$ dominating set in $3r+1$ rounds on any class of bounded expansion. Combining this algorithm, e.g., with the constant factor approximation algorithm for dominating sets on planar graphs of Lenzen et al.\ gives a constant factor approximation algorithm for connected dominating sets on planar graphs in a constant number of rounds in the $\mathcal{LOCAL}$ model, where the approximation ratio is only $6$ times larger than that of Lenzen et al.’s algorithm.

💡 Research Summary

This paper presents the first constant‑factor approximation algorithms for the distance‑r dominating set (r‑DS) and its connected variant (connected distance‑r dominating set, CD‑r‑DS) on the broad class of bounded‑expansion graphs. Bounded‑expansion graphs include planar graphs, graphs excluding a (topological) minor, and many real‑world sparse networks, and they are characterized by a uniform bound on weak coloring numbers wcol_r(G).

The authors first show how to compute, in the CONGEST_BC model, an r‑neighborhood cover of radius 2r and overlap f(r) in O(r²·log n) communication rounds. The construction builds on a distributed implementation of the routing scheme of Nešetřil and Ossona de Mendez, which itself repeatedly applies the Barenboim‑Elkin low‑outdegree orientation algorithm. The resulting cover has the property that each vertex belongs to only O(f(r)) clusters and each cluster induces a subgraph of diameter at most 2r.

Using this sparse cover, the algorithm selects one representative vertex per cluster, forming a set D. Because every vertex’s r‑neighborhood is fully contained in some cluster, D is an r‑dominating set. The selection process is purely local within each cluster, so the overall round complexity remains O(r²·log n). Moreover, the size of D is at most a constant factor of the optimum, yielding a constant‑factor approximation for r‑DS on any bounded‑expansion class.

To obtain a connected solution, the paper switches to the LOCAL model, where messages may be arbitrarily large. Starting from D, each vertex in D grows a BFS tree of depth r. In 3r + 1 LOCAL rounds the algorithm merges these trees into a single connected subgraph. The final connected set has size at most c(r)·|D|, where c(r) depends only on r and the graph class. Consequently, a constant‑factor approximation for CD‑r‑DS is achieved in O(r²·log n) CONGEST_BC rounds plus 3r + 1 LOCAL rounds.

When combined with the planar‑graph constant‑factor approximation for ordinary dominating sets by Lenzen et al., the method yields a constant‑factor approximation for connected dominating sets on planar graphs in a constant number of LOCAL rounds, with an approximation ratio at most six times that of Lenzen et al.’s algorithm. The same approach extends to bounded‑genus graphs.

The paper also surveys related work, emphasizing that prior distributed algorithms either required polylogarithmic rounds, could not guarantee constant‑factor approximations for distance‑r domination, or were limited to very specific graph families. By exploiting the structural sparsity captured by weak coloring numbers, the authors overcome known lower bounds (Ω(√log n) and Ω(log Δ) for constant‑factor domination) for the much larger family of bounded‑expansion graphs.

Overall, the work contributes a novel distributed technique for constructing sparse neighborhood covers, a clean reduction from r‑DS to CD‑r‑DS in the LOCAL model, and the first constant‑factor distributed approximations for these problems on the most general subgraph‑closed sparse graph classes known to date. Future directions include extending the approach to dynamic networks, improving the dependence on r, and exploring other distributed problems that can benefit from sparse cover constructions.