ETH Tight Algorithms for Geometric Intersection Graphs: Now in Polynomial Space

De Berg et al. in [SICOMP 2020] gave an algorithmic framework for subexponential algorithms on geometric graphs with tight (up to ETH) running times. This framework is based on dynamic programming on graphs of weighted treewidth resulting in algorithms that use super-polynomial space. We introduce the notion of weighted treedepth and use it to refine the framework of de Berg et al. for obtaining polynomial space (with tight running times) on geometric graphs. As a result, we prove that for any fixed dimension $d \ge 2$ on intersection graphs of similarly-sized fat objects many well-known graph problems including Independent Set, $r$-Dominating Set for constant $r$, Cycle Cover, Hamiltonian Cycle, Hamiltonian Path, Steiner Tree, Connected Vertex Cover, Feedback Vertex Set, and (Connected) Odd Cycle Transversal are solvable in time $2^{O(n^{1-1/d})}$ and within polynomial space.