기초과학연구원 이산수학그룹
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. …
In , 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 …
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, …
The planar separator theorem by Lipton and Tarjan 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 …