An Exact Algorithm for Optimal Areal Positioning Problem with Rectangular Targets and Requests

Abstract

In this thesis, we introduce a new class of problems, which we call Optimal Areal Positioning (OAP), and study a special form of these problems. OAPs have important applications in earth observation satellite management, tele-robotics, multi-camera control, and surveillance. In OAP, we would like to find the optimal position of a set of floating geometric objects (targets) on a two-dimensional plane to (partially) cover another set of fixed geometric objects (requests) in order to maximize the total reward obtained from covered parts of requests. In this thesis, we consider the special form of OAP in which targets and requests are parallel axes rectangles and targets are of equal size. A predetermined reward is associated with covering an area unit of each request. Based on the number of target rectangles, we classify rectangular OAP into two categories: Single Target Problem (STP) and Multi-Target Problem (MTP). The structure of MTP can be compared to the planar p-center which is NP-complete, if p is part of the input. In fact, we conjecture that MTP is NP-complete. The existing literature does not contain any work on MTP. The research contributions of this thesis are as follows: We develop new theoretical properties for the solution of STP and devised a new solution approach for it. This approach is based on a novel branch-and-bound (BB) algorithm devised over a reduced solution space. Branching is done using a clustering scheme. Our computational results show that in many cases our approach significantly outperforms the existing Plateau Vertex Traversal and brute force algorithms, especially for problems with many requests appearing in clusters over a large region. We perform a theoretical study of MTP for the first time and prove several theoretical properties for its solution. We have introduced a reduced solution space using these properties. We present the first exact algorithm to solve MTP. This algorithm has a branch-and-bound framework. The reduced solution space calls for a novel branching strategy for MTP. The algorithm has a main branch-and-bound tree with a special structure along with two trees (one for each axis) to store the information required for branching in the main tree in an efficient format. Branching is done using a clustering scheme. We perform computational experiments to evaluate the performance of our algorithm. Our algorithm solves relatively large instances of MTP in a short time