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

TL;DR

This paper refines existing algorithms for geometric intersection graphs to achieve tight, subexponential running times while using only polynomial space, applicable to many classic graph problems in fixed dimensions.

Contribution

It introduces weighted treedepth to adapt the framework for polynomial space algorithms with tight running times on geometric graphs.

Findings

Algorithms run in 2^{O(n^{1-1/d})} time for fixed dimension d.

Many graph problems are solvable within polynomial space.

Results apply to intersection graphs of fat objects in fixed dimensions.

Abstract

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.