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.
