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This paper is part of an ongoing endeavor to bring the theory of fair division closer to practice by handling requirements from real-life applications. We focus on two requirements originating from the division of land estates: (1) each agent should receive a plot of a usable geometric shape, and (2) plots of different agents must be physically separated. With these requirements, the classic fairness notion of \emph{proportionality} is impractical, since it may be impossible to attain any multiplicative approximation of it. In contrast, the \emph{ordinal maximin share approximation}, introduced by Budish in 2011, provides meaningful fairness guarantees. We prove upper and lower bounds on achievable maximin share guarantees when the usable shapes are squares, fat rectangles, or arbitrary axis-aligned rectangles, and explore the algorithmic and query complexity of finding fair partitions in this setting. Our work makes use of tools and concepts from computational geometry such as independent sets of rectangles and guillotine partitions. Appears in the 30th International Joint Conference on Artificial Intelligence (IJCAI), 2021

Semi-online models where decisions may be revoked in a limited way have been studied extensively in the last years. This is motivated by the fact that the pure online model is often too restrictive to model real-world applications, where some changes might be allowed. A well-studied measure of the amount of decisions that can be revoked is the migration factor $\beta$: When an object $o$ of size $s(o)$ arrives, the decisions for objects of total size at most $\beta\cdot s(o)$ may be revoked. Usually $\beta$ should be a constant. This means that a small object only leads to small changes. This measure has been successfully investigated for different, classic problems such as bin packing or makespan minimization. The dual of makespan minimization - the Santa Claus or machine covering problem - has also been studied, whereas the dual of bin packing - the bin covering problem - has not been looked at from such a perspective. In this work, we extensively study the bin covering problem with migration in different scenarios. We develop algorithms both for the static case - where only insertions are allowed - and for the dynamic case, where items may also depart. We also develop lower bounds for these scenarios both for amortized migration and for worst-case migration showing that our algorithms have nearly optimal migration factor and asymptotic competitive ratio (up to an arbitrary small $\eps$). We therefore resolve the competitiveness of the bin covering problem with migration.

Real-life parallel machine scheduling problems can be characterized by: (i) limited information about the exact task duration at scheduling time, and (ii) an opportunity to reschedule the remaining tasks each time a task processing is completed and a machine becomes idle. Robust optimization is the natural methodology to cope with the first characteristic of duration uncertainty, yet the existing literature on robust scheduling does not explicitly consider the second characteristic - the possibility to adjust decisions as more information about the tasks' duration becomes available, despite that re-optimizing the schedule every time new information emerges is standard practice. In this paper, we develop a scheduling approach that takes into account, at the beginning of the planning horizon, the possibility that scheduling decisions can be adjusted. We demonstrate that the suggested approach can lead to better here-and-now decisions and better makespan guarantees. To that end, we develop the first mixed integer linear programming model for adjustable robust scheduling, and a scalable two-stage approximation heuristic, where we minimize the worst-case makespan. Using this model, we show via a numerical study that adjustable scheduling leads to solutions with better and more stable makespan realizations compared to static approaches.

In a paper from 1997, Shelah asked whether $Pr_1(\lambda^+,\lambda^+,\lambda^+,\lambda)$ holds for every inaccessible cardinal $\lambda$. Here, we prove that an affirmative answer follows from $\square(\lambda^+)$. Furthermore, we establish that for every pair $\chi<\kappa$ of regular uncountable cardinals, $\square(\kappa)$ implies $Pr_1(\kappa,\kappa,\kappa,\chi)$.

We prove a quantitative variant of a disjointness theorem of nilflows from horospherical flows following a technique of Venkatesh, combined with the structural theorems for nilflows by Green, Tao and Ziegler.

COVID-19 has infected more than 68 million people worldwide since it was first detected about a year ago. Machine learning time series models have been implemented to forecast COVID-19 infections. In this paper, we develop time series models for the Gulf Cooperation Council (GCC) countries using the public COVID-19 dataset from Johns Hopkins. The dataset set includes the one-year cumulative COVID-19 cases between 22/01/2020 to 22/01/2021. We developed different models for the countries under study based on the spatial distribution of the infection data. Our experimental results show that the developed models can forecast COVID-19 infections with high precision. 9 pages, Proceedings of the 13th International Conference on Computer Modeling and Simulation, ICCMS 2021, Autoregressive integrated moving average, ARIMA, Coronavirus, COVID-19, Damped Trend, Holt Linear Trend, Machine learning, Pandemic, Time series

Stochastic Gradient Algorithms (SGAs) are ubiquitous in computational statistics, machine learning and optimisation. Recent years have brought an influx of interest in SGAs, and the non-asymptotic analysis of their bias is by now well-developed. However, relatively little is known about the optimal choice of the random approximation (e.g mini-batching) of the gradient in SGAs as this relies on the analysis of the variance and is problem specific. While there have been numerous attempts to reduce the variance of SGAs, these typically exploit a particular structure of the sampled distribution by requiring a priori knowledge of its density's mode. It is thus unclear how to adapt such algorithms to non-log-concave settings. In this paper, we construct a Multi-index Antithetic Stochastic Gradient Algorithm (MASGA) whose implementation is independent of the structure of the target measure and which achieves performance on par with Monte Carlo estimators that have access to unbiased samples from the distribution of interest. In other words, MASGA is an optimal estimator from the mean square error-computational cost perspective within the class of Monte Carlo estimators. We prove this fact rigorously for log-concave settings and verify it numerically for some examples where the log-concavity assumption is not satisfied. Comment: 51 pages, 8 figures. Revised version: an improved introduction, a completely new numerical section including experiments in non-convex settings, a new appendix discussing the dependence of the variance of SGLD on the mini-batch size

Goal-oriented semantic communication will be a pillar of next-generation wireless networks. Despite significant recent efforts in this area, most prior works are focused on specific data types (e.g., image or audio), and they ignore the goal and effectiveness aspects of semantic transmissions. In contrast, in this paper, a holistic goal-oriented semantic communication framework is proposed to enable a speaker and a listener to cooperatively execute a set of sequential tasks in a dynamic environment. A common language based on a hierarchical belief set is proposed to enable semantic communications between speaker and listener. The speaker, acting as an observer of the environment, utilizes the beliefs to transmit an initial description of its observation (called event) to the listener. The listener is then able to infer on the transmitted description and complete it by adding related beliefs to the transmitted beliefs of the speaker. As such, the listener reconstructs the observed event based on the completed description, and it then takes appropriate action in the environment based on the reconstructed event. An optimization problem is defined to determine the perfect and abstract description of the events while minimizing the transmission and inference costs with constraints on the task execution time and belief efficiency. Then, a novel bottom-up curriculum learning (CL) framework based on reinforcement learning is proposed to solve the optimization problem and enable the speaker and listener to gradually identify the structure of the belief set and the perfect and abstract description of the events. Simulation results show that the proposed CL method outperforms traditional RL in terms of convergence time, task execution cost and time, reliability, and belief efficiency.

Projection-free optimization via different variants of the Frank-Wolfe (FW) method has become one of the cornerstones in large scale optimization for machine learning and computational statistics. Numerous applications within these fields involve the minimization of functions with self-concordance like properties. Such generalized self-concordant (GSC) functions do not necessarily feature a Lipschitz continuous gradient, nor are they strongly convex. Indeed, in a number of applications, e.g. inverse covariance estimation or distance-weighted discrimination problems in support vector machines, the loss is given by a GSC function having unbounded curvature, implying absence of theoretical guarantees for the existing FW methods. This paper closes this apparent gap in the literature by developing provably convergent FW algorithms with standard O(1/k) convergence rate guarantees. If the problem formulation allows the efficient construction of a local linear minimization oracle, we develop a FW method with linear convergence rate. Comment: This is an extended version of the conference paper arXiv:2002.04320

SN~2018aoz is a Type Ia SN with a $B$-band plateau and excess emission in the infant-phase light curves $\lesssim$ 1 day after first light, evidencing an over-density of surface iron-peak elements as shown in our previous study. Here, we advance the constraints on the nature and origin of SN~2018aoz based on its evolution until the nebular phase. Near-peak spectroscopic features show the SN is intermediate between two subtypes of normal Type Ia: Core-Normal and Broad-Line. The excess emission could have contributions from the radioactive decay of surface iron-peak elements as well as ejecta interaction with either the binary companion or a small torus of circumstellar material. Nebular-phase limits on H$\alpha$ and He~I favour a white dwarf companion, consistent with the small companion size constrained by the low early SN luminosity, while the absence of [O~I] and He~I disfavours a violent merger of the progenitor. Of the two main explosion mechanisms proposed to explain the distribution of surface iron-peak elements in SN~2018aoz, the asymmetric Chandrasekhar-mass explosion is less consistent with the progenitor constraints and the observed blueshifts of nebular-phase [Fe~II] and [Ni~II]. The helium-shell double-detonation explosion is compatible with the observed lack of C spectral features, but current 1-D models are incompatible with the infant-phase excess emission, $B_{\rm max}-V_{\rm max}$ color, and absence of nebular-phase [Ca~II]. Although the explosion processes of SN~2018aoz still need to be more precisely understood, the same processes could produce a significant fraction of Type Ia SNe that appear normal after $\sim$ 1 day. Comment: Submitted for publication in ApJ. 35 pages, 16 figures, 7 tables