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 $2^{O(n^{1-1/d})}$ 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.

翻译：最近的研究表明，许多经典图问题——独立集、支配集、哈密顿回路等——在单位球图上可在亚指数时间内求解。更精确地说，在$\mathbb R^d$中的单位球图上，这些问题可在$2^{O(n^{1-1/d})}$时间内求解，该结果在指数时间假设（ETH）下是紧的。该结论可推广至相似尺寸胖物体交图的场景。对于独立集问题，相同时间复杂度可推广至非相似尺寸胖物体及其加权版本。本文证明此类推广对于支配集问题很可能不可行：基于ETH假设，我们证明——对于$\mathbb R^3$中（非单位）球图的支配集问题，不存在$2^{o(n)}$时间复杂度的算法；对于$\mathbb R^3$中单位球图的加权支配集问题，不存在$2^{o(n)}$时间复杂度的算法；对于平面上任意凸体（但具有非恒定复杂度）交图的支配集、连通支配集或斯坦纳树问题，均不存在$2^{o(n)}$时间复杂度的算法。