Fuzzy relation equations (II): the branch-point-solutions and the categorized minimal solutions

This paper presents some novel theoretical results as well as practical algorithms and computational procedures on fuzzy relation equations (FRE). These results refine and improve what has already been reported in a significant manner. In the previous paper, the authors have already proved that the problem of solving the system of fuzzy relation equations is an NP-hard problem. Therefore, it is practically impossible to determine all minimal solutions for a large system if P not equivalent to NP. In this paper, an existence theorem is proven: there exists a special branch-point-solution that is greater than all minimal solutions and less than the maximum solution. Such branch-point-solution can be calculated based on the solution-base-matrix. Furthermore, a procedure for determining all branch-point-solutions is designed. We also provide efficient algorithms which is capable of determining as well as searching for certain types of minimal solutions. We have thus obtained: (1) a fast algorithm to determine whether a solution is a minimal solution, (2) the algorithm to search for the minimal solutions that has at least a minimum value at a component in the solution vector, and (3) the procedure of determining if a system of fuzzy relation equations has the unique minimal solution. Other properties are also investigated.