A Framework for ETHTight Algorithms and Lower Bounds in Geometric Intersection Graphs
Abstract
We give an algorithmic and lowerbound framework that facilitates the construction of subexponential algorithms and matching conditional complexity bounds. It can be applied to intersection graphs of similarlysized fat objects, yielding algorithms with running time $2^{O(n^{11/d})}$ 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 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 ddimensional grids, and it allows us to derive matching $2^{\Omega(n^{11/d})}$ lower bounds under the Exponential Time Hypothesis even in the much more restricted class of $d$dimensional induced grid graphs.
 Publication:

arXiv eprints
 Pub Date:
 March 2018
 arXiv:
 arXiv:1803.10633
 Bibcode:
 2018arXiv180310633D
 Keywords:

 Computer Science  Computational Geometry;
 Computer Science  Data Structures and Algorithms
 EPrint:
 41 pages. v4 corrects a small mistake in the conference version of Theorem 1 by slightly restricting its scope and adding Lemma 4 to its proof