A framework for exponential-time-hypothesis-tight algorithms and lower bounds in geometric intersection graphs

We give an algorithmic and lower bound framework that facilitates the construction of subexponential algorithms and matching conditional complexity bounds. It can be applied to intersection graphs of similarly-sized fat objects, yielding algorithms with running time 2O(n1-1/d) for any fixed dimension d ≥ 2 for many well-known graph problems, including Independent Set, r-Dominating Set for constant r, and Steiner Tree. For most problems, we get improved running times compared to prior work; in some cases, we give the first known subexponential algorithm in geometric intersection graphs. Additionally, most of the obtained algorithms are representationagnostic, i.e., they work on the graph itself and do not require the geometric representation. Our algorithmic framework is based on a weighted separator theorem and various treewidth techniques. The lower bound framework is based on a constructive embedding of graphs into d-dimensional grids, and it allows us to derive matching 2Ω (n1-1/d) lower bounds under the exponential time hypothesis even in the much more restricted class of d-dimensional induced grid graphs.