KAIST Discrete Math Seminar

Imagine you want to present your collection of n coins on a shelf, taking as little space as possible – how should you arrange the coins?
More precisely, we are given n circular disks of different radii, and we want to place them in the plane so that they touch the x-axis from above, such that no two disks overlap. The goal is to minimize the length of the range from the leftmost point on a disk to the rightmost point on a disk.
On this seemingly innocent problem we will meet a wide range of algorithmic concepts: An efficient algorithm for a special case, an NP-hardness proof, an approximation algorithm with a guaranteed approximation factor, APX-hardness, and a quasi-polynomial time approximation scheme.

We prove a generalization of Pal’s 1921 conjecture that if a convex shape P can be placed in any orientation inside a convex shape Q in the plane, then P can also be turned continuously through 360 degrees inside Q. We also prove a lower bound of Ω(m n²) on the number of combinatorially distinct maximal placements of a convex m-gon P in a convex n-gon Q. This matches the upper bound proven by Agarwal et al.

Packing problems are concerned with positioning geometric objects so that they do not overlap and require an amount of space as small as possible. They have been investigated within mathematics for centuries starting with the famous Kepler conjecture. There are many surprising properties and open problems in connection with packing. The lecture will give a short survey about these “classical” issues but then concentrate on algorithms for packing. Since already the most simple variants are NP-hard, it makes sense to look for efficient approximation algorithms. We will present constant factor approximations for packing rectangles and convex polygons into containers which are minimum area rectangles or convex sets. Algorithms for analogous problems concerning three-dimensional objects will be presented, as well.
This is joint research with Mark de Berg, Christian Knauer, Léonard von Niederhäusern, and Nadja Scharf.

A well known theorem of Grötzsch states that every planar graph is 3-colorable. We will show a simple proof based on a recent result of Kostochka and Yancey on the number of edges in 4-critical graphs. Then we show a strengthening of the Grötzsch’s theorem in several different directions. Based on joint works with Ilkyoo Choi, Jan Ekstein, Zdeněk Dvořák, Přemek Holub, Alexandr Kostochka, and Matthew Yancey.

In this talk, for a fixed positive integer d at least 3, we study upper
bounds for the second largest eigenvalue in (an n-vertex) d-regular graph to
guarantee a certain vertex-connectivity.