Prophet Inequalities for Cost Minimization

Prophet inequalities for rewards maximization are fundamental to optimal stopping theory with several applications to mechanism design and online optimization. We study the cost minimization counterpart of the classical prophet inequality, where one is facing a sequence of costs $X_1, X_2, \dots, X_n$ in an online manner and must stop at some point and take the last cost seen. Given that the $X_i$'s are independent, drawn from known distributions, the goal is to devise a stopping strategy $S$ that minimizes the expected cost. If the $X_i$'s are not identically distributed, then no strategy can achieve a bounded approximation if the arrival order is adversarial or random. This leads us to consider the case where the $X_i$'s are I.I.D.. For the I.I.D. case, we give a complete characterization of the optimal stopping strategy, and show that, if our distribution satisfies a mild condition, then the optimal stopping strategy achieves a tight (distribution-dependent) constant-factor approximation. Our techniques provide a novel approach to analyze prophet inequalities, utilizing the hazard rate of the distribution. We also show that when the hazard rate is monotonically increasing (i.e. the distribution is MHR), this constant is at most $2$, and this is optimal for MHR distributions. For the classical prophet inequality, single-threshold strategies can achieve the optimal approximation factor. Motivated by this, we analyze single-threshold strategies for the cost prophet inequality problem. We design a threshold that achieves a $\operatorname{O}\left(\operatorname{polylog}{n}\right)$-factor approximation, where the exponent in the logarithmic factor is a distribution-dependent constant, and we show a matching lower bound. We note that our results can be used to design approximately optimal posted price-style mechanisms for procurement auctions which may be of independent interest.Comment: 38 page

Extraction of reflectance maps for smart farming applications using Unmanned Aerial Vehicles

Summarization: In this application paper, a robust framework for smart remote sensing of cultivations using Unmanned Aerial Vehicles is presented, yielding to a useful tool with advanced capabilities in terms of time-efficiency, accuracy, user-friendly operability, adjustability and expandability. The proposed system incorporates multispectral imaging, automated navigation and real-time monitoring functionalities into a fixed-wing Unmanned Aerial Vehicle platform. Offline analysis of captured data is performed, at this stage of system development, via powerful commercial software so as to extract the reflection map of the crop area under study based on the Normalized Difference Vegetation Index. The proposed approach has been tested on selected cultivations in two regions (Greece), aiming at recording field variability and early detecting factors related to crop stress. Preliminary results indicate that the proposed framework can prove a cost-effective, precise, flexible and operative solution for agriculture industry, enabling the application of smart farming procedures for productive farm management. Adopting a collaborative group of aerial vehicles via Flying Ad hoc Networks, the proposed sensing approach could be further enhanced for large-scale applications, fusing data from multiple nodes into an advanced Decision Support System and providing information on bigger areas at the same time with respect to a single sensing source.Presented on