arXiv subjects

Minki Kim

Publications and source records attributed to Minki Kim.

34 records · Page 2Linked to original sources

arXiv2021

A system of disjoint representatives of line segments with given $k$ directions

We prove that for all positive integers $n$ and $k$, there exists an integer $N = N(n,k)$ satisfying the following. If $U$ is a set of $k$ direction vectors in the plane and $\mathcal{J}_U$ is the set of all line segments in direction $u$ for some $u\in U$, then for every $N$ families $\mathcal{F}_1, \ldots, \mathcal{F}_N$, each consisting of $n$ mutually disjoint segments in $\mathcal{J}_U$, there is a set $\{A_1, \ldots, A_n\}$ of $n$ disjoint segments in $\bigcup_{1\leq i\leq N}\mathcal{F}_i$ and distinct integers $p_1, \ldots, p_n\in \{1, \ldots, N\}$ satisfying that $A_j\in \mathcal{F}_{p_j}$ for all $j\in \{1, \ldots, n\}$. We generalize this property for underlying lines on fixed $k$ directions to $k$ families of simple curves with certain conditions.

math.CO↗

arXiv2021

Domination numbers and noncover complexes of hypergraphs

Let $\mathcal{H}$ be a hypergraph on a finite set $V$. A {\em cover} of $\mathcal{H}$ is a set of vertices that meets all edges of $\mathcal{H}$. If $W$ is not a cover of $\mathcal{H}$, then $W$ is said to be a {\em noncover} of $\mathcal{H}$. The {\em noncover complex} of $\mathcal{H}$ is the abstract simplicial complex whose faces are the noncovers of $\mathcal{H}$. In this paper, we study homological properties of noncover complexes of hypergraphs. In particular, we obtain an upper bound on their Leray numbers. The bound is in terms of hypergraph domination numbers. Also, our proof idea is applied to compute the homotopy type of the noncover complexes of certain uniform hypergraphs, called {\em tight paths} and {\em tight cycles}. This extends to hypergraphs known results on graphs.

math.CO↗

arXiv2020

End-to-End Evaluation of Federated Learning and Split Learning for Internet of Things

This work is the first attempt to evaluate and compare felderated learning (FL) and split neural networks (SplitNN) in real-world IoT settings in terms of learning performance and device implementation overhead. We consider a variety of datasets, different model architectures, multiple clients, and various performance metrics. For learning performance, which is specified by the model accuracy and convergence speed metrics, we empirically evaluate both FL and SplitNN under different types of data distributions such as imbalanced and non-independent and identically distributed (non-IID) data. We show that the learning performance of SplitNN is better than FL under an imbalanced data distribution, but worse than FL under an extreme non-IID data distribution. For implementation overhead, we end-to-end mount both FL and SplitNN on Raspberry Pis, and comprehensively evaluate overheads including training time, communication overhead under the real LAN setting, power consumption and memory usage. Our key observations are that under IoT scenario where the communication traffic is the main concern, the FL appears to perform better over SplitNN because FL has the significantly lower communication overhead compared with SplitNN, which empirically corroborate previous statistical analysis. In addition, we reveal several unrecognized limitations about SplitNN, forming the basis for future research.

cs.CR↗

arXiv2020

The layer number of $α$-evenly distributed point sets

For a finite point set in $\mathbb{R}^d$, we consider a peeling process where the vertices of the convex hull are removed at each step. The layer number $L(X)$ of a given point set $X$ is defined as the number of steps of the peeling process in order to delete all points in $X$. It is known that if $X$ is a set of random points in $\mathbb{R}^d$, then the expectation of $L(X)$ is $Θ(|X|^{2/(d+1)})$, and recently it was shown that if $X$ is a point set of the square grid on the plane, then $L(X)=Θ(|X|^{2/3})$. In this paper, we investigate the layer number of $α$-evenly distributed point sets for $α>1$; these point sets share the regularity aspect of random point sets but in a more general setting. The set of lattice points is also an $α$-evenly distributed point set for some $α>1$. We find an upper bound of $O(|X|^{3/4})$ for the layer number of an $α$-evenly distributed point set $X$ in a unit disk on the plane for some $α>1$, and provide an explicit construction that shows the growth rate of this upper bound cannot be improved. In addition, we give an upper bound of $O(|X|^{\frac{d+1}{2d}})$ for the layer number of an $α$-evenly distributed point set $X$ in a unit ball in $\mathbb{R}^d$ for some $α>1$ and $d\geq 3$.

math.CO↗

arXiv2020

Can We Use Split Learning on 1D CNN Models for Privacy Preserving Training?

A new collaborative learning, called split learning, was recently introduced, aiming to protect user data privacy without revealing raw input data to a server. It collaboratively runs a deep neural network model where the model is split into two parts, one for the client and the other for the server. Therefore, the server has no direct access to raw data processed at the client. Until now, the split learning is believed to be a promising approach to protect the client's raw data; for example, the client's data was protected in healthcare image applications using 2D convolutional neural network (CNN) models. However, it is still unclear whether the split learning can be applied to other deep learning models, in particular, 1D CNN. In this paper, we examine whether split learning can be used to perform privacy-preserving training for 1D CNN models. To answer this, we first design and implement an 1D CNN model under split learning and validate its efficacy in detecting heart abnormalities using medical ECG data. We observed that the 1D CNN model under split learning can achieve the same accuracy of 98.9\% like the original (non-split) model. However, our evaluation demonstrates that split learning may fail to protect the raw data privacy on 1D CNN models. To address the observed privacy leakage in split learning, we adopt two privacy leakage mitigation techniques: 1) adding more hidden layers to the client side and 2) applying differential privacy. Although those mitigation techniques are helpful in reducing privacy leakage, they have a significant impact on model accuracy. Hence, based on those results, we conclude that split learning alone would not be sufficient to maintain the confidentiality of raw sequential data in 1D CNN models.

cs.CR↗

arXiv2019

Complexes of graphs with bounded independence number

Let $G=(V,E)$ be a graph and $n$ a positive integer. Let $I_n(G)$ be the abstract simplicial complex whose simplices are the subsets of $V$ that do not contain an independent set of size $n$ in $G$. We study the collapsibility numbers of the complexes $I_n(G)$ for various classes of graphs, focusing on the class of graphs with maximum degree bounded by $Δ$. As an application, we obtain the following result: Let $G$ be a claw-free graph with maximum degree at most $Δ$. Then, every collection of $\left\lfloor\left(\fracΔ{2}+1\right)(n-1)\right\rfloor+1$ independent sets in $G$ has a rainbow independent set of size $n$.

math.CO↗

arXiv2019

Rainbow independent sets in certain classes of graphs

For a given class $\mathcal{C}$ of graphs and given integers $m \leq n$, let $f_\mathcal{C}(n,m)$ be the minimal number $k$ such that every $k$ independent $n$-sets in any graph belonging to $\mathcal{C}$ have a (possibly partial) rainbow independent $m$-set. Motivated by known results on the finiteness and actual value of $f_\mathcal{C}(n,m)$ when $\mathcal{C}$ is the class of line graphs of graphs, we study this function for various other classes.

math.CO↗

arXiv2019

Word-representability of Toeplitz graphs

math.CO↗

arXiv2019

Choice functions in the intersection of matroids

We prove a common generalization of two results, one on rainbow fractional matchings and one on rainbow sets in the intersection of two matroids: Given $d = r \lceil k \rceil - r + 1$ functions of size (=sum of values) $k$ that are all independent in each of $r$ given matroids, there exists a rainbow set of $supp(f_i)$, $i \leq d$, supporting a function with the same properties.

math.CO↗

arXiv2018

On operations preserving semi-transitive orientability of graphs

We consider the class of semi-transitively orientable graphs, which is a much larger class of graphs compared to transitively orientable graphs, in other words, comparability graphs. Ever since the concept of a semi-transitive orientation was defined as a crucial ingredient of the characterization of alternation graphs, also knownas word-representable graphs, it has sparked independent interest. In this paper, we investigate graph operations and graph products that preserve semitransitive orientability of graphs. The main theme of this paper is to determine which graph operations satisfy the following statement: if a graph operation is possible on a semitransitively orientable graph, then the same graph operation can be executed on the graph while preserving the semi-transitive orientability. We were able to prove that this statement is true for edge-deletions, edge-additions, and edge-liftings. Moreover, for all three graph operations,we showthat the initial semi-transitive orientation can be extended to the new graph obtained by the graph operation. Also, Kitaev and Lozin explicitly asked if certain graph products preserve the semitransitive orientability. We answer their question in the negative for the tensor product, lexicographic product, and strong product.We also push the investigation further and initiate the study of sufficient conditions that guarantee a certain graph operation to preserve the semi-transitive orientability.

math.CO↗

arXiv2018

On k-11-representable graphs

Distinct letters $x$ and $y$ alternate in a word $w$ if after deleting in $w$ all letters but the copies of $x$ and $y$ we either obtain a word of the form $xyxy\cdots$ (of even or odd length) or a word of the form $yxyx\cdots$ (of even or odd length). A simple graph $G=(V,E)$ is word-representable if there exists a word $w$ over the alphabet $V$ such that letters $x$ and $y$ alternate in $w$ if and only if $xy$ is an edge in $E$. Thus, edges of $G$ are defined by avoiding the consecutive pattern 11 in a word representing $G$, that is, by avoiding $xx$ and $yy$. In 2017, Jeff Remmel has introduced the notion of a $k$-$11$-representable graph for a non-negative integer $k$, which generalizes the notion of a word-representable graph. Under this representation, edges of $G$ are defined by containing at most $k$ occurrences of the consecutive pattern $11$ in a word representing $G$. Thus, word-representable graphs are precisely $0$-$11$-representable graphs. Our key result in this paper is showing that any graph is $2$-$11$-representable by a concatenation of permutations, which is rather surprising taking into account that concatenation of permutations has limited power in the case of $0$-$11$-representation. Also, we show that the class of word-representable graphs, studied intensively in the literature, is contained strictly in the class of $1$-$11$-representable graphs. Another result that we prove is the fact that the class of interval graphs is precisely the class of $1$-$11$-representable graphs that can be represented by uniform words containing two copies of each letter. This result can be compared with the known fact that the class of circle graphs is precisely the class of $0$-$11$-representable graphs that can be represented by uniform words containing two copies of each letter.

math.CO↗

arXiv2018

Nerves, minors, and piercing numbers

We make the first step towards a "nerve theorem" for graphs. Let $G$ be a simple graph and let $\mathcal{F}$ be a family of induced subgraphs of $G$ such that the intersection of any members of $\mathcal{F}$ is either empty or connected. We show that if the nerve complex of $\mathcal{F}$ has non-vanishing homology in dimension three, then $G$ contains the complete graph on five vertices as a minor. As a consequence we confirm a conjecture of Goaoc concerning an extension of the planar $(p,q)$ theorem due to Alon and Kleitman: Let $\mathcal{F}$ be a finite family of open connected sets in the plane such that the intersection of any members of $\mathcal{F}$ is either empty or connected. If among any $p \geq 3$ members of $\mathcal{F}$ there are some three that intersect, then there is a set of $C$ points which intersects every member of $\mathcal{F}$, where $C$ is a constant depending only on $p$.

math.CO↗

arXiv2017

Notes on complexity of packing coloring

A packing $k$-coloring for some integer $k$ of a graph $G=(V,E)$ is a mapping $φ:V\to\{1,\ldots,k\}$ such that any two vertices $u, v$ of color $φ(u)=φ(v)$ are in distance at least $φ(u)+1$. This concept is motivated by frequency assignment problems. The \emph{packing chromatic number} of $G$ is the smallest $k$ such that there exists a packing $k$-coloring of $G$. Fiala and Golovach showed that determining the packing chromatic number for chordal graphs is \NP-complete for diameter exactly 5. While the problem is easy to solve for diameter 2, we show \NP-completeness for any diameter at least 3. Our reduction also shows that the packing chromatic number is hard to approximate within $n^{{1/2}-\varepsilon}$ for any $\varepsilon > 0$. In addition, we design an \FPT algorithm for interval graphs of bounded diameter. This leads us to exploring the problem of finding a partial coloring that maximizes the number of colored vertices.

cs.CC↗

arXiv2016

On tiling the integers with $4$-sets of the same gap sequence

Partitioning a set into similar, if not, identical, parts is a fundamental research topic in combinatorics. The question of partitioning the integers in various ways has been considered throughout history. Given a set $\{x_1, \ldots, x_n\}$ of integers where $x_1<\cdots<x_n$, let the {\it gap sequence} of this set be the nondecreasing sequence $d_1, \ldots, d_{n-1}$ where $\{d_1, \ldots, d_{n-1}\}$ equals $\{x_{i+1}-x_i:i\in\{1,\ldots, n-1\}\}$ as a multiset. This paper addresses the following question, which was explicitly asked by Nakamigawa: can the set of integers be partitioned into sets with the same gap sequence? The question is known to be true for any set where the gap sequence has length at most two. This paper provides evidence that the question is true when the gap sequence has length three. Namely, we prove that given positive integers $p$ and $q$, there is a positive integer $r_0$ such that for all $r\geq r_0$, the set of integers can be partitioned into $4$-sets with gap sequence $p, q$, $r$.

math.CO↗

arXiv2015

A note on the colorful fractional Helly theorem

Helly's theorem is a classical result concerning the intersection patterns of convex sets in $\mathbb{R}^d$. Two important generalizations are the colorful version and the fractional version. Recently, Bárány et al. combined the two, obtaining a colorful fractional Helly theorem. In this paper, we give an improved version of their result.

math.CO↗

arXiv2008

Secrecy Capacity over Correlated Ergodic Fading Channel

We investigate the secrecy capacity of an ergodic fading wiretap channel in which the main channel is correlated with the eavesdropper channel. In this study, the full Channel State Information (CSI) is assumed, and thus the transmitter knows the channel gains of the legitimate receiver and the eavesdropper. By analyzing the resulting secrecy capacity we quantify the loss of the secrecy capacity due to the correlation. In addition, we study the asymptotic behavior of the secrecy capacity as Signal-to-Noise Ratio (SNR) tends to infinity. The capacity of an ordinary fading channel logarithmically increases with SNR. On the contrary, the secrecy capacity converges into a limit which can be an upper bound on the secrecy capacity over the fading wiretap channel. We find a closed form of the upper bound for the correlated Rayleigh wiretap channel which also includes the independent case as a special one. Our work shows that the upper bound is determined by only two channel parameters; the correlation coefficient and the ratio of the main to the eavesdropper channel gains that will be called PCC and CGR respectively. The analysis of the upper bound tells how the two channel parameters affect the secrecy capacity and leads to the conclusion that the excessively large signal power does not provide any advantage in the secrecy capacity, and the loss due to the correlation is especially serious in low CGR regime.

cs.IT↗