Robust Optimization and Groundwork for Problem Mapping

Goals of this research were to develop a conceptual algorithm that can optimize execution time for generating a solution set and demonstrate that a solution set of one sub-problem can be applied to another sub-problem within the same problem set. To achieve the proposed goals, GloPro was developed to generate rule sets for different sub-problems within a problem set, as well as identifying which rule sets are to be utilized for a given instance of the problem. The algorithm was to be robust, as to be applicable to a wide array of problems without radical re-design per problem. This idea was fueled by the concept of Structure-Mapping Theory, where a set of knowledge is mapped from one domain to another based on the shared baseline characteristics. Utilizing a Genetic Algorithm (GA), plus A* with a classifier hybrid, the algorithm includes a period of supervised learning followed by execution in an operational environment. Progressive learning occurred through application of the algorithm to multiple sub-problems, each having unique characteristics. The algorithm was applied to a simulated robotic agent in a maze environment as a proxy for other problems. This problem is well known, but still an active problem in the field of robotics. The experimental results indicate that the hybrid GA with A* technique is feasible, and that progressive learning is enhanced through application of previous learning results to a period of learning. In addition, the evolved solutions were unique to the sub-problems, indicating that this technique can be used to develop robust solutions across sub-problems

The Difficulty of Approximating the Chromatic Number for Random Composite Graphs

Combinatorial Optimization is an important class of techniques for solving Combinatorial Problems. Many practical problems are Combinatorial Problems, such as the Traveling Salesman Problem (TSP) and Composite Graph Coloring Problem (CGCP). Unfortunately, both of these problems are NP-complete and it is not known if efficient algorithms exist to solve these problems. Even approximation with guaranteed results can be just as difficult. Recently, many generalized search techniques have been developed to improve upon the solutions found by the heuristic algorithms. This paper presents results for CGCP. In particular, exact and heuristic algorithms are presented and analyzed. This study is made, to show empirically that CGCP cannot provide guarantees on the approximation using these heuristic methods. In addition, an improvement is presented on the interchange method by Clementson and Elphick that is used with vertex sequential algorithms. This improvement allows graphs of up to 1000 vertices to be colored in considerably less time than previous studies. The study also shows that CDSaturl heuristic does not compete as well with CDSatur as expected for large graphs with edge density of 0.2. Several NP-completeness theorems are presented and proved. Approximation of CGCP is shown to be as difficult as finding exact solutions if we expect the approximate solutions to fall within a specified bound. These bounds on approximate solutions are shown to be directly related to the bounds that have been proved to exist for the Standard Graph Coloring Problem (SGCP). Finally, a model of CGCP is developed so that the Tabu Search technique can be applied. Several neighborhoods are developed and tested on 50 and 100 vertex graphs. Timing and performance is analyzed against the heuristics in the previous study. Instances of larger order graphs are used to test the best neighborhood searches with Tabu Search

LIPIcs, Volume 261, ICALP 2023, Complete Volume

LIPIcs, Volume 261, ICALP 2023, Complete Volum

LIPIcs, Volume 258, SoCG 2023, Complete Volume

LIPIcs, Volume 258, SoCG 2023, Complete Volum