Approximation of k-Dominating Set below exhaustive-search time

Determine whether k-Dominating Set admits a $(\log n)^{1-o(1)}$-approximation algorithm running in $n^{k-0.1}$ time, where $n$ is the number of vertices in the input graph.

Background

The paper situates its results within the broader problem of understanding the tradeoff between approximation quality and running time for parameterized covering problems. A survey by Feldmann, Karthik, Lee, and Manurangsi formulates the cited question for $k$-Dominating Set: whether one can obtain an approximation factor nearly logarithmic in the graph size while saving a fixed constant from the exhaustive-search exponent $k$.

The results in the paper establish near-logarithmic running-time lower bounds for parameterized Set Cover, including a lower bound under SETH against algorithms with running time $O(|\Gamma|^{k-\delta})$. However, the explicitly cited question concerns $k$-Dominating Set, so it is stated as a separate unresolved question in the paper’s discussion of prior work.