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We give an algorithmic and lower-bound framework that facilitates theconstruction of subexponential algorithms and matching conditional complexitybounds. It can be applied to intersection graphs of similarly-sized fatobjects, yielding algorithms with running time $2^{O(n^{1-1/d})}$ for any fixeddimension $d \geq 2$ for many well known graph problems, including IndependentSet, $r$-Dominating Set for constant $r$, and Steiner Tree. For most problems,we get improved running times compared to prior work; in some cases, we givethe first known subexponential algorithm in geometric intersection graphs.Additionally, most of the obtained algorithms are representation-agnostic,i.e., they work on the graph itself and do not require the geometricrepresentation. Our algorithmic framework is based on a weighted separatortheorem and various treewidth techniques.  The lower bound framework is based on a constructive embedding of graphs intod-dimensional grids, and it allows us to derive matching$2^{\Omega(n^{1-1/d})}$ lower bounds under the Exponential Time Hypothesis evenin the much more restricted class of $d$-dimensional induced grid graphs.

A Framework for ETH-Tight Algorithms and Lower Bounds in Geometric Intersection Graphs