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

by   Mark de Berg, et al.
Budapest University of Technology and Economics
Utrecht University
TU Eindhoven

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 a wide range of geometric intersection graphs (intersections of similarly sized fat objects), yielding algorithms with running time 2^O(n^1-1/d) for any fixed dimension d ≥ 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 work on the graph itself, i.e., do not require any geometric information. 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^Ω(n^1-1/d) lower bounds under the Exponential Time Hypothesis even in the much more restricted class of d-dimensional induced grid graphs.


page 1

page 2

page 3

page 4


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

We give an algorithmic and lower-bound framework that facilitates the co...

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

De Berg et al. in [SICOMP 2020] gave an algorithmic framework for subexp...

Uniting General-Graph and Geometric-Based Radio Networks via Independence Number Parametrization

In the study of radio networks, the tasks of broadcasting (propagating a...

Optimality Program in Segment and String Graphs

Planar graphs are known to allow subexponential algorithms running in ti...

Hyperbolic intersection graphs and (quasi)-polynomial time

We study unit ball graphs (and, more generally, so-called noisy uniform ...

How does object fatness impact the complexity of packing in d dimensions?

Packing is a classical problem where one is given a set of subsets of Eu...

A Framework for Robust Realistic Geometric Computations

We propose a new paradigm for robust geometric computations that complem...

Please sign up or login with your details

Forgot password? Click here to reset