Discrete Mathematics Group

Daniel Mock, A Simple for the Dominating Set Problem and More
2026-01-06 16:30 - 17:30 (Asia/Seoul) | Room B332, IBS (기초과학연구원)
In [1], Fabianski et. al. developed a simple, yet surprisingly powerful algorithmic framework to develop efficient parameterized graph algorithms. Notably they derive a simple parameterized algorithm for the dominating set problem on a variety of graph classes, including powers of nowhere dense classes and biclique-free classes. These results encompass a wide range of previously known results and often improve the best known running times. Similar results follow for the distance-r variation of dominating set and for independent set. The running time of the algorithm is closely tied to model-theoretic properties, i.e. stability and the Helly property.
We build upon these results and develop a similar algorithm which only relies on the strong Helly property and does not need stability. For the dominating set problem, we get a parameterized algorithm that works (additionally to biclique-free classes and powers of nowhere dense classes) weakly gamma-closed classes while being simpler and faster than previously known results.
In this talk, we introduce the basic framework, results by Fabianski et. al and connections to other areas. We discuss our new insights and possible research directions.
[1] Grzegorz Fabianski, Michal Pilipczuk, Sebastian Siebertz, Szymon Torunczyk: Progressive Algorithms for Domination and Independence. STACS 2019

Ferdinand Ihringer, Boolean Functions Analysis in the Grassmann Graph
2026-01-13 16:30 - 17:30
Boolean function analysis for the hypercube ${ 0, 1 }^n$ is a well-developed field and has many famous results such as the FKN Theorem or Nisan-Szegedy Theorem. One easy example is the classification of Boolean degree $1$ functions: If $f$ is a real, $n$-variate affine function which is Boolean on the $n$-dimensional hypercube (that is, $f(x) in { 0, 1 }$ for $x in { 0, 1 }^n$), then $f(x) = 0$, $f(x) = 1$, $f(x) = x_i$ or $f(x) = 1 – x_i$. The same classification (essentially) holds if we restrict ${ 0, 1}^n$ to elements with Hamming weight $k$ if $n-k, k geq 2$. If we replace $k$-sets of ${ 1, ldots, n }$ by $k$-spaces in $V(n, q)$, the $n$-dimensional vector space over the field with $q$ elements, then suddenly even the simple question of classifying Boolean degree $1$ functions, here traditionally known as Cameron-Liebler classes, becomes seemingly hard to solve.
We will discuss some results on low-degree Boolean functions in the vector space setting. Most notably, we will discuss how vector space Ramsey numbers, so extremal combinatorics, can be utilized in this finite geometrical setting.

Tomáš Masařík, Separator Theorem for Minor-free Graphs in Linear Time
2026-01-20 16:30 - 17:30
The planar separator theorem by Lipton and Tarjan [FOCS ’77, SIAM Journal on Applied Mathematics ’79] states that any planar graph with $n$ vertices has a balanced separator of size $O(sqrt{n})$ that can be found in linear time. This landmark result kicked off decades of research on designing linear or nearly linear-time algorithms on planar graphs. In an attempt to generalize Lipton-Tarjan’s theorem to nonplanar graphs, Alon, Seymour, and Thomas [STOC ’90, Journal of the AMS ’90] showed that any minor-free graph admits a balanced separator of size $O(sqrt{n})$ that can be found in $O(n^{3/2})$ time. The superlinear running time in their separator theorem is a key bottleneck for generalizing algorithmic results from planar to minor-free graphs. Despite extensive research for more than two decades, finding a balanced separator of size $O(sqrt{n})$ in (linear) $O(n)$ time for minor-free graphs remains a major open problem. Known algorithms either give a separator of size much larger than $O(sqrt{n})$ or have superlinear running time, or both.
In this paper, we answer the open problem affirmatively. Our algorithm is very simple: it runs a vertex-weighted variant of breadth-first search (BFS) a constant number of times on the input graph. Our key technical contribution is a weighting scheme on the vertices to guide the search for a balanced separator, offering a new connection between the size of a balanced separator and the existence of a clique-minor model. We believe that our weighting scheme may be of independent interest.
This is a joint work with Édouard Bonnet, Tuukka Korhonen, Hung Le, and Jason Li.

Daniel Dadush, TBA
2026-01-27 16:30 - 17:30

Seonghun Park (박성훈), TBA
2026-02-10 16:30 - 17:30

Marek Sokołowski, TBA
2026-02-24 16:30 - 17:30

Chính T. Hoàng, TBA
2026-03-03 16:30 - 17:30

Dario Cavallaro, TBA
2026-03-10 16:30 - 17:30

Hidde Koerts, TBA
2026-03-24 16:30 - 17:30

Tung H. Nguyen, TBA
2026-03-31 16:30 - 17:30

Xavier Goaoc, TBA
2026-05-19 16:30 - 17:30

Maria Chudnovsky, TBA
2026-05-26 16:30 - 17:30
