Upper Domination: Complexity and Approximation.

We consider Upper Domination, the problem of finding a maximum cardinality minimal dominating set in a graph. We show that this problem does not admit an \(n^{1-\epsilon }\) approximation for any \(\epsilon >0\), making it significantly harder than Dominating Set, while it remains hard even on severely restricted special cases, such as cubic graphs (APX-hard), and planar subcubic graphs (NP-hard). We complement our negative results by showing that the problem admits an \(O(\varDelta )\) approximation on graphs of maximum degree \(\varDelta \), as well as an EPTAS on planar graphs. Along the way, we also derive essentially tight \(n^{1-\frac{1}{d}}\) upper and lower bounds on the approximability of the related problem Maximum Minimal Hitting Set on d-uniform hypergraphs, generalising known results for Maximum Minimal Vertex Cover.