Executive Summary : | A fundamental assumption in classical algorithmic and geometric research is to consider the input data to be precise. This assumption is not justified in reality. The real-world data in general is prone to be uncertain in terms of location, labeling, or some other parameter due to inadequate reachability, rounding-off data, heterogeneous data sources, and so on. In the literature, such a data set is commonly known as an uncertain or imprecise data set. Over the past few decades, various interesting paradigms have been developed to cope up with the uncertain data to make the algorithmic techniques more relevant to practical applications. Different models are also proposed to represent this uncertainty in data as different geometric objects (such as line, disks, squares, etc) for each data point under a suitable distance metric. These geometric objects are known as imprecise regions or regions of uncertainty. Facility Location problem is another versatile field of study, which stems from its variety of applications in the real-world. A wide range of objectives and mathematical formulations lead to different variations of the problem, in theoretical computer science specially in computational geometry. These problems originate from real-world scenarios, lead to different researches that seek optimal or near-optimal solutions while determining the computational complexity of the problem. In a general formulation, there are different types of facilities $F$ available, with each facility having potentially multiple copies. A distance function $\partial$ is given. Our objective is to determine the values of certain decision variables (e.g., to select the ``best'' facility or set of facilities) to optimize the objective function. Given a set ${\cal R}=\{R_1,R_2.\ldots, R_n\}$ of $n$ regions and $k$ colors, each region is assigned to exactly one of $k$ possible colors. A colored point set ${\cal P}=\{p_1,p_2,\ldots,p_n\}$ is called as a realization of the set of regions ${\cal R}$, if there exists a color-preserving one-to-one correspondence between ${\cal P}$ and ${\cal R}$, and each point $p_i \in {\cal P}$ is contained inside the corresponding region $R_i\in {\cal R}$. The general goal of a problem under this framework, is to find an appropriate realization of $\cal R$ that optimizes the objective function. The problem specific objectives are mentioned below. |