Heterochromatic Higher Order Transversals for Convex Sets

In this short paper, we show that if $\left\{\mathcal{F}_{n}\right\}_{n \in \mathbb{N}}$ be a collection of families compact $(r, R)$-fat convex sets in $\mathbb{R}^{d}$ and if every heterochromatic sequence with respect to $\left\{\mathcal{F}_{n}\right\}_{n \in \mathbb{N}}$ contains $k+2$ convex sets that can be pierced by a $k$-flat then there exists a family $\mathcal{F}_{m}$ from the collection that can be pierced by finitely many $k$-flats. Additionally, we show that if $\left\{\mathcal{F}_{n}\right\}_{n \in \mathbb{N}}$ be a collection of families of compact convex sets in $\mathbb{R}^{d}$ where each $\mathcal{F}_{n}$ is a family of closed balls (axis parallel boxes) in $\mathbb{R}^{d}$ and every heterochromatic sequence with respect to $\left\{\mathcal{F}_{n}\right\}_{n \in \mathbb{N}}$ contains $2$ intersecting closed balls (boxes) then there exists a family $\mathcal{F}_{m}$ from the collection that can be pierced by a finite number of points from $\mathbb{R}^{d}$. To complement the above results, we also establish some impossibility of proving similar results for other more general families of convex sets. Our results are a generalization of $(\aleph_0,k+2)$-Theorem for $k$-transversals of convex sets by Keller and Perles (Symposium on Computational Geometry 2022), and can also be seen as a colorful infinite variant of $(p,q)$-Theorems of Alon and Klietman (Advances in Mathematics 1992), and Alon and Kalai (Discrete & Computational Geometry 1995).Comment: 16 pages and 5 figures. Section 3 rewritte

A Dynamic Weighted Federated Learning for Android Malware Classification

Android malware attacks are increasing daily at a tremendous volume, making Android users more vulnerable to cyber-attacks. Researchers have developed many machine learning (ML)/ deep learning (DL) techniques to detect and mitigate android malware attacks. However, due to technological advancement, there is a rise in android mobile devices. Furthermore, the devices are geographically dispersed, resulting in distributed data. In such scenario, traditional ML/DL techniques are infeasible since all of these approaches require the data to be kept in a central system; this may provide a problem for user privacy because of the massive proliferation of Android mobile devices; putting the data in a central system creates an overhead. Also, the traditional ML/DL-based android malware classification techniques are not scalable. Researchers have proposed federated learning (FL) based android malware classification system to solve the privacy preservation and scalability with high classification performance. In traditional FL, Federated Averaging (FedAvg) is utilized to construct the global model at each round by merging all of the local models obtained from all of the customers that participated in the FL. However, the conventional FedAvg has a disadvantage: if one poor-performing local model is included in global model development for each round, it may result in an under-performing global model. Because FedAvg favors all local models equally when averaging. To address this issue, our main objective in this work is to design a dynamic weighted federated averaging (DW-FedAvg) strategy in which the weights for each local model are automatically updated based on their performance at the client. The DW-FedAvg is evaluated using four popular benchmark datasets, Melgenome, Drebin, Kronodroid and Tuandromd used in android malware classification research.Comment: Accepted in SoCTA 202

Almost covering all the layers of hypercube with multiplicities

Given a hypercube $\mathcal{Q}^{n} := \{0,1\}^{n}$ in $\mathbb{R}^{n}$ and $k \in \{0, \dots, n\}$, the $k$-th layer $\mathcal{Q}^{n}_{k}$ of $\mathcal{Q}^{n}$ denotes the set of all points in $\mathcal{Q}^{n}$ whose coordinates contain exactly $k$ many ones. For a fixed $t \in \mathbb{N}$ and $k \in \{0, \dots, n\}$, let $P \in \mathbb{R}\left[x_{1}, \dots, x_{n}\right]$ be a polynomial that has zeroes of multiplicity at least $t$ at all points of $\mathcal{Q}^{n} \setminus \mathcal{Q}^{n}_{k}$, and $P$ has zeros of multiplicity exactly $t-1$ at all points of $\mathcal{Q}^{n}_{k}$. In this short note, we show that $$deg(P) \geq \max\left\{ k, n-k\right\}+2t-2.$$Matching the above lower bound we give an explicit construction of a family of hyperplanes $H_{1}, \dots, H_{m}$ in $\mathbb{R}^{n}$, where $m = \max\left\{ k, n-k\right\}+2t-2$, such that every point of $\mathcal{Q}^{n}_{k}$ will be covered exactly $t-1$ times, and every other point of $\mathcal{Q}^{n}$ will be covered at least $t$ times. Note that putting $k = 0$ and $t=1$, we recover the much celebrated covering result of Alon and F\"uredi (European Journal of Combinatorics, 1993). Using the above family of hyperplanes we disprove a conjecture of Venkitesh (The Electronic Journal of Combinatorics, 2022) on exactly covering symmetric subsets of hypercube $\mathcal{Q}^{n}$ with hyperplanes. To prove the above results we have introduced a new measure of complexity of a subset of the hypercube called index complexity which we believe will be of independent interest. We also study a new interesting variant of the restricted sumset problem motivated by the ideas behind the proof of the above result.Comment: 16 pages, substantial changes from previous version, title and abstract changed to better reflect the content of the pape

Dimension Independent Helly Theorem for Lines and Flats

We give a generalization of dimension independent Helly Theorem of Adiprasito, B\'{a}r\'{a}ny, Mustafa, and Terpai (Discrete & Computational Geometry 2022) to higher dimensional transversal. We also prove some impossibility results that establish the tightness of our extension.Comment: 10 page

Stabbing boxes with finitely many axis-parallel lines and flats

We give necessary and sufficient condition for an infinite collection of axis-parallel boxes in $\mathbb{R}^{d}$ to be pierceable by finitely many axis-parallel $k$-flats, where $0 \leq k < d$. We also consider colorful generalizations of the above result and establish their feasibility. The problem considered in this paper is an infinite variant of the Hadwiger-Debrunner $(p,q)$-problem.Comment: 13 page

On higher multiplicity hyperplane and polynomial covers for symmetry preserving subsets of the hypercube

Alon and F\"uredi (European J. Combin. 1993) gave a tight bound for the following hyperplane covering problem: find the minimum number of hyperplanes required to cover all points of the n-dimensional hypercube {0,1}^n except the origin. Their proof is among the early instances of the polynomial method, which considers a natural polynomial (a product of linear factors) associated to the hyperplane arrangement, and gives a lower bound on its degree, whilst being oblivious to the (product) structure of the polynomial. Thus, their proof gives a lower bound for a weaker polynomial covering problem, and it turns out that this bound is tight for the stronger hyperplane covering problem. In a similar vein, solutions to some other hyperplane covering problems were obtained, via solutions of corresponding weaker polynomial covering problems, in some special cases in the works of the fourth author (Electron. J. Combin. 2022), and the first three authors (Discrete Math. 2023). In this work, we build on these and solve a hyperplane covering problem for general symmetric sets of the hypercube, where we consider hyperplane covers with higher multiplicities. We see that even in this generality, it is enough to solve the corresponding polynomial covering problem. Further, this seems to be the limit of this approach as far as covering symmetry preserving subsets of the hypercube is concerned. We gather evidence for this by considering the class of blockwise symmetric sets of the hypercube (which is a strictly larger class than symmetric sets), and note that the same proof technique seems to only solve the polynomial covering problem

State or market? How to effectively decrease alcohol-related crash fatalities and injuries

Background It is estimated that more than 270 000 people die yearly in alcohol-related crashes globally. To tackle this burden, government interventions, such as laws which restrict blood alcohol concentration (BAC) levels and increase penalties for drunk drivers, have been implemented. The introduction of private-sector measures, such as ridesharing, is regarded as alternatives to reduce drunk driving and related sequelae. However, it is unclear whether state and private efforts complement each other to reduce this public health challenge. Methods We conducted interrupted time-series analyses using weekly alcohol-related traffic fatalities and injuries per 1 000 000 population in three urban conglomerates (Santiago, Valparaíso and Concepción) in Chile for the period 2010–2017. We selected cities in which two state interventions—the ‘zero tolerance law’ (ZTL), which decreased BAC, and the ‘Emilia law’ (EL), which increased penalties for drunk drivers—were implemented to decrease alcohol-related crashes, and where Uber ridesharing was launched. Results In Santiago, the ZTL was associated with a 29.1% decrease (95% CI 1.2 to 70.2), the EL with a 41.0% decrease (95% CI 5.5 to 93.2) and Uber with a non-significant 28.0% decrease (95% CI −6.4 to 78.5) in the level of weekly alcohol-related traffic fatalities and injuries per 1 000 000 population series. In Concepción, the EL was associated with a 28.9% reduction (95% CI 4.3 to 62.7) in the level of the same outcome. In Valparaíso, the ZTL had a −0.01 decrease (95% CI −0.02 to −0.00) in the trend of weekly alcohol-related crashes per 1 000 000 population series. Conclusion In Chile, concomitant decreases of alcohol-related crashes were observed after two state interventions were implemented but not with the introduction of Uber. Relationships between public policy interventions, ridesharing and motor vehicle alcohol-related crashes differ between cities and over time, which might reflect differences in specific local characteristics