기초과학연구원 이산수학그룹
Vertex Cover is perhaps the most-studied problem in parameterized complexity that frequently serves as a testing ground for new concepts and techniques. In this talk, I will focus on a generalization of Vertex Cover called Component Order Connectivity (COC). Given a graph G, an integer k and a positive integer d, the task is to …
We consider a continuous model of graphs, introduced by Dearing and Francis in 1974, where each edge of G to be a unit interval, giving rise to an infinite metric space that contains not only the vertices of G but all points on all edges of G. Several standard graph problems can be defined and …
Given a $k$-uniform hypergraph $F$, its Turán density $\pi(F)$ is the infimum over all $d\in $ such that any $n$-vertex $k$-uniform hypergraph $H$ with at least $d\binom{n}{k}+o(n^k)$ edges contains a copy of $F$. While Turán densities are generally well understood for graphs ($k=2$), the problem becomes notoriously difficult for $k\geq 3$, even for small hypergraphs. …
Consider a general Turan-type problem on hypergraphs. Let $\mathcal{F}$ be a family of $k$-subsets of $$ that does not contain sets $F_1, \ldots, F_s$ satisfying some property $P$. We show that if $P$ is low-dimensional in some sense (e.g., is defined by intersections of bounded size) then, under polynomial dependencies between $n, k$ and the …
Let $F_{k,d}(n)$ be the maximal size of a set ${A}\subseteq $ such that the equation \ has no solution with $a_1,a_2,\ldots,a_k\in A$ and integer $x$. Erdős, Sárközy and T. Sós studied $F_{k,2}$, and gave bounds when $k=2,3,4,6$ and also in the general case. We study the problem for $d=3$, and provide bounds for $k=2,3,4,6$ and …
Let A be a random n×n matrix with independent entries, and suppose that the entries are “uniformly anticoncentrated” (for example, A could be a uniformly random n×n matrix with ±1 entries). We prove that the permanent of A is exponentially anticoncentrated, significantly improving previous bounds of Tao and Vu. Our proof also works for the …
We present a dynamic data structure that maintains a tree decomposition of width at most 9k+8 of a dynamic graph with treewidth at most k, which is updated by edge insertions and deletions. The amortized update time of our data structure is $2^{O(k)} \log n$, where n is the number of vertices. The data structure …
Given a prime power $q \equiv 1 \pmod 4$, the Paley graph of order $q$ is the graph defined over $\mathbb{F}_q$ (the finite field with $q$ elements), such that two vertices are adjacent if and only if their difference is a square in $\mathbb{F}_q$. In this talk, I will present some recent progress on the …
We present a new cryptomorphic definition of orthogonal matroids with coefficients using Grassmann-Plücker functions. The equivalence is motivated by Cayley's identities expressing principal and almost-principal minors of a skew-symmetric matrix in terms of its pfaffians. As a corollary of the new cryptomorphism, we deduce that each component of the orthogonal Grassmannian is parameterized by certain …
Since the development of the first randomized polynomial-time algorithm for volume computation by Dyer, Frieze, and Kannan in 1989, convex-body sampling has been a central problem at the intersection of algorithms, geometry, and probability. A major milestone came in 1997, when Kannan, Lovász, and Simonovits analyzed the Ball Walk and formulated the influential KLS conjecture. …