기초과학연구원 이산수학그룹
Given a graph, how do we construct a surface so that the graph embeds in that surface in an optimal way? Thomassen showed that for minimum genus as optimality criterion, this problem would be NP-hard. Instead of minimum genus, here we use local planarity -- and provide a polynomial algorithm. Our embedding method is based …
A rooted digraph is a vertex-flame if for every vertex v there is a set of internally disjoint directed paths from the root to v whose set of terminal edges covers all ingoing edges of v. It was shown by Lovász that every finite rooted digraph admits a spanning subdigraph which is a vertex-flame and large, where the latter means …
Many problems that are NP-hard in general become tractable on `structurally recursive’ graph classes. For example, consider classes of bounded tree- or clique-width. Since the 1990s, many directed analogues of tree-width have been proposed. However, many natural problems (e.g. directed HamiltonPath and MaxCut) remain intractable on such digraph classes of `bounded width’. In this talk, …
A convex lattice polytope is the convex hull of a set of integral points. Vershik conjectured the existence of a limit shape for random convex lattice polygons, and three proofs of this conjecture were given in the 1990s by Bárány, by Vershik, and by Sinai. To state this old result more precisely, there is a …
Given a minor-closed graph class ${\cal G}$, the (minor) obstruction of ${\cal G}$ is the set of all minor-minimal graphs not in ${\cal G}$. Given a non-negative integer $k$, we define the $k$-apex of ${\cal A}$ as the class containing every graph $G$ with a set $S$ of vertices whose removal from $G$ gives a graph …
Let $\mathbb{F}_q$ be a finite field of order $q$ which is a prime power. In the finite field setting, we say that a function $\phi\colon \mathbb{F}_q^d\times \mathbb{F}_q^d\to \mathbb{F}_q$ is a Mattila-Sjölin type function in $\mathbb{F}_q^d$ if for any $E\subset \mathbb{F}_q^d$ with $|E|\gg q^{\frac{d}{2}}$, we have $\phi(E, E)=\mathbb{F}_q$. The main purpose of this talk is to present …
Recently, significant progress has been made in the area of machine learning algorithms, and they have quickly become some of the most exciting tools in a scientist’s toolbox. In particular, recent advances in the field of reinforcement learning have led computers to reach superhuman level play in Atari games and Go, purely through self-play. In …
An intersection digraph is a digraph where every vertex $v$ is represented by an ordered pair $(S_v, T_v)$ of sets such that there is an edge from $v$ to $w$ if and only if $S_v$ and $T_w$ intersect. An intersection digraph is reflexive if $S_v\cap T_v\neq \emptyset$ for every vertex $v$. Compared to well-known undirected …
An interval graph is the intersection graph of a family of intervals in the real line. Motivated by problems in ecology, Roberts defined the boxicity of a graph G to be the minimal k such that G can be written as the intersection of k interval graphs. A natural higher-dimensional generalization of interval graphs is …
The preservation of symmetry is one of the key tools for designing data-efficient neural networks. A representative example is convolutional neural networks (CNNs); they preserve translation symmetries, and this symmetry preservation is often attributed to their success in real-world applications. In the machine-learning community, there is a growing body of work that explores a new …