Panos Giannopoulos

Parameterized Complexity of Geometric Problems

This paper surveys parameterized complexity results for hard geometric algorithmic problems. It includes fixed-parameter tractable problems in graph drawing, geometric graphs, geometric covering and several other areas, together with an overview of the algorithmic techniques used. Fixed-parameter intractability results are surveyed as well. Finally, we give some directions for future research.

Finding the best shortcut in a geometric network

Given a Euclidean graph G in Rd with n vertices and m edges we consider the problem of adding a shortcut such that the stretch factor of the resulting graph is minimized. Currently, the fastest algorithm for computing the stretch factor of a Euclidean graph runs in O(mn+n2 log n) time, resulting in a trivial O(mn3+n4 log n) time algorithm for computing the optimal shortcut. First, we show that a simple modification yields the optimal solution in O(n4) time using O(n2) space. To reduce the running times we consider several approximation algorithms. Our main result is a (2+ε)-approximation algorithm with running time O(nm+n2(log n+1/ε3d)) using O(n2) space.

Improving the Stretch Factor of a Geometric Network by Edge Augmentation

Given a Euclidean graph

Hardness of discrepancy computation and ε-net verification in high dimension

G

COMPUTING GEOMETRIC MINIMUM-DILATION GRAPHS IS NP-HARD

in

On the parameterized complexity of d -dimensional point set pattern matching

\mathbb{R}^d

Matching point sets with respect to the earth mover’s distance

with

Maximizing the area of overlap of two unions of disks under rigid motion

n

The Complexity of Separating Points in the Plane

vertices and

The Parameterized Complexity of Some Geometric Problems in Unbounded Dimension

m