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

TL;DR

This paper introduces a unified framework for designing ETH-tight algorithms and lower bounds for geometric intersection graphs, improving existing results and providing new subexponential algorithms for various problems.

Contribution

It presents a novel algorithmic and lower-bound framework applicable to intersection graphs of fat objects, enabling subexponential algorithms and matching complexity bounds.

Findings

Achieves $2^{O(n^{1-1/d})}$ algorithms for multiple graph problems

Provides the first subexponential algorithms for some geometric intersection graph problems

Establishes matching lower bounds under ETH for d-dimensional grid graphs

Abstract

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 $2^{O(n^{1-1/d})}$ for any fixed dimension $d \geq 2$ 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 representation-agnostic, 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 $2^{\Omega(n^{1-1/d})}$ lower bounds under the Exponential Time Hypothesis even in the much more restricted class of $d$-dimensional induced grid graphs.