기초과학연구원 이산수학그룹
Transductions are a general formalism for expressing transformations of graphs (and more generally, of relational structures) in logic. We prove that a graph class C can be FO-transduced from a class of bounded-height trees (that is, has bounded shrubdepth) if, and only if, from C one cannot FO-transduce the class of all paths. This establishes …
SATNet is a differentiable constraint solver with a custom backpropagation algorithm, which can be used as a layer in a deep-learning system. It is a promising proposal for bridging deep learning and logical reasoning. In fact, SATNet has been successfully applied to learn, among others, the rules of a complex logical puzzle, such as Sudoku, …
We show that if $L_1$ and $L_2$ are linear transformations from $\mathbb{Z}^d$ to $\mathbb{Z}^d$ satisfying certain mild conditions, then, for any finite subset $A$ of $\mathbb{Z}^d$, \ This result corrects and confirms the two-summand case of a conjecture of Bukh and is best possible up to the lower-order term for many choices of $L_1$ and …
One of the important work in graph theory is the graph minor theory developed by Robertson and Seymour in 1980-2010. This provides a complete description of the class of graphs that do not contain a fixed graph H as a minor. Later on, several generalizations of H-minor free graphs, which are sparse, have been defined …
Twin-width is a recently introduced graph parameter based on vertex contraction sequences. On classes of bounded twin-width, problems expressible in FO logic can be solved in FPT time when provided with a sequence witnessing the bound. Classes of bounded twin-width are very diverse, notably including bounded rank-width, $\Omega ( \log (n) )$-subdivisions of graphs of …
The sphere packing problem asks for the densest packing of nonoverlapping equal-sized balls in the space. This is an old and difficult problem in discrete geometry. In this talk, we give a new proof for the result that for $ 1<p<2 $, the translative packing density of superballs (a generalization of $\ell^p$ balls) in $\mathbb{R}^n$ …
Given a set $E$ and a point $y$ in a vector space over a finite field, the radial projection $\pi_y(E)$ of $E$ from $y$ is the set of lines that through $y$ and points of $E$. Clearly, $|\pi_y(E)|$ is at most the minimum of the number of lines through $y$ and $|E|$. I will discuss …
One interesting difference between (nondegenerate) Graph Turán problem and Hypergraph Turán problem is that the hypergraph families can have at least two very different extremal constructions. In this talk, we will look at some recent progress and approaches to constructing hypergraph families with at least two different extremal constructions. Based on some joint work with …
This talk will highlight recent results establishing a beautiful computational phase transition for approximate counting/sampling in (binary) undirected graphical models (such as the Ising model or on weighted independent sets). The computational problem is to sample from the equilibrium distribution of the model or equivalently approximate the corresponding normalizing factor known as the partition function. We show that when correlations die …
A hole in a graph $G$ is an induced cycle of length at least four, and for every hole $H$ in $G$, a vertex $h\in G\setminus H$ is called a $t$-hub for $H$ if $h$ has at least $t$ neighbor in $H$. Sintiari and Trotignon were the first to construct graphs with arbitrarily large treewidth …