Lower Bounds for Dominating Set in Ball Graphs and for Weighted Dominating Set in Unit-Ball Graphs

Recently it was shown that many classic graph problems – Independent Set, Dominating Set, Hamiltonian Cycle, and more – can be solved in subexponential time on unit-ball graphs. More precisely, these problems can be solved in $2^{O(n^{1-1\mathrm{/}d})}$ time on unit-ball graphs in ${\mathbb{R}}^d$, which is tight under ETH. The result can be generalized to intersection graphs of similarly-sized fat objects. For Independent Set the same running time can be achieved for non-similarly-sized fat objects, and for the weighted version of the problem. We show that such generalizations most likely are not possible for Dominating Set: assuming ETH, we prove that - there is no algorithm with running time $2^{o(n)}$ for Dominating Set on (non-unit) ball graphs in ${\mathbb{R}}^3$; - there is no algorithm with running time $2^{o(n)}$ for Weighted Dominating Set on unit-ball graphs in ${\mathbb{R}}^3$; - there is no algorithm with running time $2^{o(n)}$ for Dominating Set, Connected Dominating Set, or Steiner Tree on intersections graphs of arbitrary convex (but non-constant-complexity) objects in the plane.