## Theory of Combinatorial Algorithms

Prof. Emo Welzl and Prof. Bernd Gärtner

# Mittagsseminar (in cooperation with A. Steger, D. Steurer and B. Sudakov)

 Mittagsseminar Talk Information

Date and Time: Tuesday, October 17, 2017, 12:15 pm

Duration: 30 minutes

Location: OAT S15/S16/S17

Speaker: Luis Barba

## Convex hulls in the presence of obstacles

We study generalizations of convex hulls to polygonal domains with holes. Convexity in Euclidean space is based on the notion of shortest paths, which are straight-line segments. In a polygonal domain, shortest paths are polygonal paths called geodesics. One possible generalization of convex hull is based on the rubber band'' conception of the convex hull as a shortest curve that encloses a given set of sites. However, we show that it is NP-hard to compute such a curve in a general polygonal domain. Hence we focus on a different, more direct generalization of convexity, where a set $X$ is geodesically convex if it contains all geodesics between every pair of points $x,y\in X$. The corresponding geodesic convex hull presents a few surprises and turns out to behave quite differently compared to the classic Euclidean setting and to the geodesic hull inside a simple polygon. As a main result, we describe a class of polygonal domains that suffice to represent geodesic convex hulls and characterize which such domains are geodesically convex. Using such a representation we present an algorithm to construct the geodesic convex hull of a set of $O(n)$ sites in a polygonal domain with a total of $n$ vertices and $h$ holes in $O(n^3h^{3+\varepsilon})$ time.

This is joint work with M. Hoffmann, M. Korman and A. Pilz

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