Several classes of polynomial-time solvable fuzzy relational inequalities

Finding the list of all minimal solutions of a fuzzy relational system is a tough work. Actually it has been proved to be NP hard recently by Markovskii. (A.V. Markovskii, On the relation between equations with max-product composition and the covering problem. Fuzzy Sets Systems 153, pp. 261-273, 2005). However, practical programs for solving these problems usually run much faster than they are supposed to be in theoretical result. This motivates us to ask: are there any polynomial-time solvable fuzzy relation systems and what kind of systems they should be? This paper devotes to answering this question by analyzing the computational complexity of a proposed algorithm for solving fuzzy relation systems. It is proved that a fuzzy relation system has polynomial time algorithm whenever it has poly(m, n) many quasi-minimal solutions, where m × n is its dimension. Based on the conclusion, some conditions are proposed, under which fuzzy relation systems can be solved in polynomial time. Presented examples show the practicality of these conditions.