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 n1− approximation for any > 0, making it significantly harder than Dominating Set, while it remains hard even on severely restricted special cases, such as cubic graphs (APXhard), and planar subcubic graphs (NP-hard). We complement our negative results by showing that the problem admits an O(∆) approximation on graphs of maximum degree ∆, as well as an EPTAS on planar graphs. Along the way, we also derive essentially tight n1− 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.