With the technological advancement in computer science, Geographic Information Science (GIScience), and transportation, more and more complex path finding queries including category based queries are proposed and studied across diverse disciplines. A category based query, such as Optimal Sequenced Routing (OSR) queries and Trip Planning Queries (TPQ), asks for a minimum-cost path that traverses a set of categories with or without a predefined order in a graph. Due to the extensive computing time required to process these complex queries in a large scale environment, efficient algorithms are highly desirable whenever processing time is a consideration. In Artificial Intelligence (AI), a best first search is an informed heuristic path finding algorithm that uses domain knowledge as heuristics to expedite the search process. Traditional best first searches are single-variate in terms of the number of variables to describe a state, and thus not appropriate to process these queries in a graph. In this dissertation, 1) two new types of category based queries, Category Sequence Traversal Query (CSTQ) and Optimal Sequence Traversal Query (OSTQ), are proposed; 2) the existing single-variate best first searches are extended to multivariate best first searches in terms of the state specified, and a class of new concepts--state graph, sub state graph, sub state graph space, local heuristic, local admissibility, local consistency, global heuristic, global admissibility, and global consistency--is introduced into best first searches; 3) two bivariate best first search algorithms, C* and O*, are developed to process CSTQ and OSTQ in a graph, respectively; 4) for each of C* and O*, theorems on optimality and optimal efficiency in a sub state graph space are developed and identified; 5) a family of algorithms including C*-P, C-Dijkstra, O*-MST, O*-SCDMST, O*- Dijkstra, and O*-Greedy is identified, and case studies are performed on path finding in transportation networks, and/or fully connected graphs, either directed or undirected; and 6) O*- SCDMST is adopted to efficiently retrieve optimal solutions for OSTQ using network distance metric in a large transportation network. / Ph. D.