Decision trees are one of the most commonly used tools in decision analysis, knowledge representation, machine learning, etc., for its simplicity and interpretability. We consider an extension of dynamic programming approach to process the whole set of decision trees for the given decision table which was previously only attainable by brute-force algorithms. We study decision tables with many-valued decisions (each row may contain multiple decisions) because they are more reasonable models of data in many cases. To address this problem in a broad sense, we consider not only decision trees but also inhibitory trees where terminal nodes are labeled with “̸= decision”. Inhibitory trees can sometimes describe more knowledge from datasets than decision trees. As for cost functions, we consider depth or average depth to minimize time complexity of trees, and the number of nodes or the number of the terminal, or nonterminal nodes to minimize the space complexity of trees. We investigate the multi-stage optimization of trees relative to some cost functions, and also the possibility to describe the whole set of strictly optimal trees. Furthermore, we study the bi-criteria optimization cost vs. cost and cost vs. uncertainty for decision trees, and cost vs. cost and cost vs. completeness for inhibitory trees. The most interesting application of the developed technique is the creation of multi-pruning and restricted multi-pruning approaches which are useful for knowledge representation and prediction. The experimental results show that decision trees constructed by these approaches can often outperform the decision trees constructed by the CART algorithm. Another application includes the comparison of 12 greedy heuristics for single- and bi-criteria optimization (cost vs. cost) of trees. We also study the three approaches (decision tables with many-valued decisions, decision tables with most common decisions, and decision tables with generalized decisions) to handle inconsistency of decision tables. We also analyze the time complexity of decision and inhibitory trees over arbitrary sets of attributes represented by information systems in the frameworks of local (when we can use in trees only attributes from problem description) and global (when we can use in trees arbitrary attributes from the information system) approaches.

This thesis is devoted to the development of extensions of dynamic programming to the study of decision trees. The considered extensions allow us to make multi-stage optimization of decision trees relative to a sequence of cost functions, to count the number of optimal trees, and to study relationships: cost vs cost and cost vs uncertainty for decision trees by construction of the set of Pareto-optimal points for the corresponding bi-criteria optimization problem. The applications include study of totally optimal (simultaneously optimal relative to a number of cost functions) decision trees for Boolean functions, improvement of bounds on complexity of decision trees for diagnosis of circuits, study of time and memory trade-off for corner point detection, study of decision rules derived from decision trees, creation of new procedure (multi-pruning) for construction of classifiers, and comparison of heuristics for decision tree construction. Part of these extensions (multi-stage optimization) was generalized to well-known combinatorial optimization problems: matrix chain multiplication, binary search trees, global sequence alignment, and optimal paths in directed graphs.

In recent years, edge computing has become an increasingly popular computing paradigm to enable real-time data processing and mobile intelligence. Edge computing allows computing at the edge of the network, where data is generated and distributed at the nearby edge servers to reduce the data access latency and improve data processing efficiency. In addition, with the advance of Artificial Intelligence of Things (AIoT), not only millions of data are generated from daily smart devices, such as smart light bulbs, smart cameras, and various sensors, but also a large number of parameters of complex machine learning models have to be trained and exchanged by these AIoT devices. Classical cloud-based platforms have difficulty communicating and processing these data/models effectively with sufficient privacy and security protection. Due to the heterogeneity of edge elements including edge servers, mobile users, data resources, and computing tasks, the key challenge is how to effectively manage resources (e.g. data, services) and schedule tasks (e.g. ML/FL tasks) in the edge clouds to meet the QoS of mobile users or maximize the platform's utility. To that end, this dissertation studies joint resource management and task scheduling for mobile edge computing. The key contributions of the dissertation are two-fold. Firstly, we study the data placement problem in edge computing and propose a popularity-based method as well as several load-balancing strategies to effectively place data in the edge network. We further investigate a joint resource placement and task dispatching problem and formulate it as an optimization problem. We propose a two-stage optimization method and a reinforcement learning (RL) method to maximize the total utilities of all tasks. Secondly, we focus on a specific computing task, i.e., federated learning (FL), and study the joint participant selection and learning scheduling problem for multi-model federated edge learning. We formulate a joint optimization problem and propose several multi-stage optimization algorithms to solve the problem. To further improve the FL performance, we leverage the power of the quantum computing (QC) technique and propose a hybrid quantum-classical Benders' decomposition (HQCBD) algorithm as well as a multiple-cuts version to accelerate the convergence speed of the HQCBD algorithm. We show that the proposed algorithms can achieve the consistent optimal value compared with the classical Benders' decomposition running in the classical CPU computer, but with fewer convergence iterations. / Computer and Information Science