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    Area of Science:

    • Mathematics
    • Computer Science
    • Operations Research

    Background:

    • Symmetry is prevalent in nature and can optimize computational efficiency by reducing redundancy.
    • Permutation-related combinatorial optimization problems often exhibit complex solution spaces.
    • Group theory provides a framework for understanding and analyzing symmetric structures.

    Purpose of the Study:

    • To explore the symmetric structure of solution spaces in permutation-related combinatorial optimization problems using group theory.
    • To develop a novel search strategy that exploits identified symmetric features to enhance computational efficiency.
    • To adapt the strategy for cases where inherent symmetry conditions are not met.

    Main Methods:

    • Application of group theory and group action to analyze the solution space of permutation problems.
    • Design of a half-solution-space search strategy applicable to common search operators.
    • Development of dimension mapping approaches to enable the strategy when symmetry conditions are not met.
    • Empirical evaluation on benchmark instances for Single Row Facility Layout Problem (SRFLP), Traveling Salesman Problem (TSP), and Multi-Objective Traveling Salesman Problem (MOTSP).

    Main Results:

    • Discovery of conditions for meaningful symmetric features based on objective function form and object number.
    • Demonstration that the half-solution-space search strategy enhances the exploration of promising regions without increased computational cost.
    • Successful application of the strategy to SRFLP, TSP, and MOTSP benchmark instances.
    • Algorithms utilizing the strategy showed more competitive performance compared to those that did not exploit symmetry.

    Conclusions:

    • The proposed half-solution-space search strategy effectively leverages symmetry in combinatorial optimization.
    • This approach offers a method to improve the performance of optimization algorithms by exploiting inherent structural properties.
    • The dimension mapping techniques extend the applicability of symmetry exploitation to a broader range of problems.