## Theory of Combinatorial Algorithms

Prof. Emo Welzl and Prof. Bernd Gärtner

# Mittagsseminar (in cooperation with M. Ghaffari, A. Steger and B. Sudakov)

 Mittagsseminar Talk Information

Date and Time: Tuesday, September 11, 2012, 11:30 am

Duration: 30 minutes

Location: CAB G51

Speaker: Gábor Tardos (Alfréd Rényi Institute of Mathematics)

## Tight lower bounds for geometric epsilon nets

Assume we are given a set $S$ of $n$ points in the plane and want find a subset $T$ of these points with the property that every half-plane containing at least $\epsilon n$ points of $S$ contains at least one point of $T$. Such a set $T$ is called an $\epsilon$-net for $S$ with respect to half-planes. The well known theorem of Haussler and Welzl states that an $\epsilon$-net of size $O((1/\epsilon)\log(1/\epsilon))$ exists for any set $S$ and not just with respect to half-planes but also with respect to any any bounded VC-dimension collection of ranges. (All reasonable collections such as triangles, pentagons, disks, etc. have bounded VC dimension.)

While the Haussler-Welzl theorem is very general in many specific geometric settings such as the half-planes or even half-spaces in 3D a stronger statement is also true: $O(1/\epsilon)$ size $\epsilon$-nets exist (Matousek, Seidel, Welzl). In other cases intermediate results were shown: Ezra and Sharir showed recently that for axis-aligned rectangles in the plane or axis-aligned boxes in 3D $O((1/\epsilon)\log\log(1/\epsilon))$ size $\epsilon$-nets exist. Based on these and similar results a "general belief" was formed that the Haussler-Welzl bound is never tight in geometric settings and in "reasonable" cases a linear (that is $O(1/\epsilon)$) bound holds.

Contrary to this belief we construct point sets $S$ in the 4 dimensional Euclidean space such that all $\epsilon$-nets with respect to either half-spaces or axis-aligned boxes have size $\Omega((1/\epsilon)\log(1/\epsilon))$ showing that the Haussler-Welzl bound is tight in this case. In another construction we find planar point sets showing that the Ezra-Sharir bound on $\epsilon$-nets with respect to axis-aligned rectangles is also tight.

Previous talks by year:   2018  2017  2016  2015  2014  2013  2012  2011  2010  2009  2008  2007  2006  2005  2004  2003  2002  2001  2000  1999  1998  1997  1996

Information for students and suggested topics for student talks