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Cover-decomposition of multiple coverings under conditions involving randomness

Periodic Reporting for period 1 - CovDecRND (Cover-decomposition of multiple coverings under conditions involving randomness)

Reporting period: 2015-09-01 to 2017-08-31

Combinatorial geometry is a very active field where most problems have real life applications. The study of multiple coverings was initiated by Davenport and L. Fejes Tóth 50 years ago. In 1986 J. Pach published the first papers about decomposability of multiple coverings. It was discovered recently that besides its theoretical interest, this area has important practical applications. Now there is a great activity in this field with several breakthrough results. The main goal of this project is to study cover-decomposability, polychromatic colorings and related notions for different geometric and abstract families of sets under various additional conditions. For an illustration of a typical question, suppose that an area is covered by disks such that every point is contained in at least m disks, where m is a large integer. Is it possible to partition the disks into two parts such that each part alone covers the whole area?

Any problem that involves partitioning into groups can be modeled through decompositions. These include many practical problems, such as job scheduling and bin packing, that have important real world applications. Because of this large diversity, there are many different questions one can ask about decompositions. The underlying relation of the objects to be partitioned can be usually described by a graph or hypergraph. These notions are general enough to capture a wide variety of problems. Ramsey-type coloring questions of graphs come up in several seemingly unrelated fields and have motivated a large part of the research in combinatorics. To achieve suitable decompositions, diverse mathematical tools have been applied, including the probabilistic method, linear algebra and topological methods. There are also numerous generalizations of the concept of partitions, including assigning vector values, graph homomorphisms and matroid theory, which achieve new results and provide deeper insight. This shows that decompositions play a central and important role in combinatorics, and in general, the whole of mathematics.

For a practical application, suppose that a given area needs to be monitored by sensors with a given location and range, a fixed part of the area for each that it can monitor. These sensors can vary significantly depending on the application, from detectors to patrols with a fixed base. Suppose further that every point of the area is in the range of several sensors that can monitor it. Then for any point this gives the possibility to share the job of monitoring it between the sensors to whose range it belongs to. Inactive sensors can save energy, used for other jobs, or activated at a later point of time, depending on the application. In our model, suppose that each sensor also has an associated lifetime for which it can remain active. This can correspond to energy or other restrictions, for example, it can be a given number of hours in case of solar powered batteries. The goal of the sensor cover problem is to create a time schedule which determines when each sensor is active so that the whole area is constantly monitored, for as long as possible. (Or in case of solar powered devices with a fixed amount of active hours per day, the goal is to determine the feasibility of maintaining the surveillance all day.) Suppose that the lifetime of each sensor is the same, and for each we need to pick a time slot during which it stays active. The different time slots will be the parts of the decomposition we are looking for.
Probably the most important part of the project has been the finalization of the work started by the PI that has refuted the 1980 main conjecture of the field that was proposed by János Pach. This has led to a joint publication of Pach and the PI that has appeared in the highly prestigious journal of Advances in Mathematics. In this work the construction of the PI has been extended to the families of translates of all smooth planar convex sets from disks, and other, positive results, have been also established.

The other important publication of the project is a joint work with Balázs Keszegh and has appeared in the proceedings of the top conference of the field, the 33rd International Symposium on Computational Geometry (SoCG 2017). In this work a new conjecture has been proposed that will hopefully motivate future researchers of the field for decades, as the 1980 conjecture of Pach did until recently. Building on an earlier result of Ackerman, Keszegh and Vizer, this new conjecture has been verified for homothetic copies of convex polygons.

The PI participated in several other publications, in total 11 works acknowledge the support of the grant, of which 7 have appeared in journals or conference proceedings, and the remaining 4 have been submitted (and are available on the arXiv repository).

The PI has attended numerous conferences and workshops, and have been invited to give talks on several seminars where the results of the project have been promoted. The most important of these perhaps has been the 2017 Colloquia in Combinatorics in London where the PI has been invited to deliver a talk on Coloring geometric hypergraphs. Another notable attended event (apart from the above mentioned Symposium on Computational Geometry that took place in Brisbane, Australia) was the Oberwolfach Workshop on Discrete Geometry where the leading experts of the field have been invited. Dissemination has also included talks invited for wider audience, such as organized by the Cambridge University Hungarian Society in Cambridge and by the Vodafone Project YOU in Budapest.

The PI has also taken several further duties in the scientific community, such as being Editor of a Special Issue of the Journal of Computational Geometry (JoCG) 7(2), Program committee member of the 34th International Symposium on Computational Geometry (SoCG 2018) and the 28th International Symposium on Algorithms and Computation (ISAAC 2017), and Organizing committee member of the aforementioned SoCG 2018.

The acknowledgment of the work performed by the PI is best shown by having been awarded a 5-year long Momentum (Lendület) grant by the Hungarian Academy of Sciences (MTA) to continue research in the field as the head of a new group in Budapest. The PI has also received a Burgen scholarship from the Academia Europaea.
It was proved for the sensor cover problem that if the sensors are ill-placed, then it is possible that every point is covered by an arbitrarily large number of sensors, yet it is not possible to find a schedule in which each sensor is active for only 12 hours a day such that the whole area is monitored at all times. (This appeared in János Pach and Dömötör Pálvölgyi. Unsplittable coverings in the plane. Advances in Mathematics, 302: 433-457, 2016.) It is, however, possible to find a schedule where each sensor is active only 18 hours a day and it is conjectured that even 16 hours a day is sufficient for any initial placement of the sensors. (This appeared in Balázs Keszegh and Dömötör Pálvölgyi. Proper Coloring of Geometric Hypergraphs. Proceedings of the 33rd International Symposium on Computational Geometry (SoCG 2017), 47:1-47:15, Leibniz International Proceedings in Informatics (LIPIcs) 77, Dagstuhl, Germany, 2017.) These theoretical results give general impossibility results and a possible framework for the design of practical algorithms that solve real life instances of the sensor cover problem and similar questions.
Summary of known results for polychromatic colorings