Solving linear objective optimization problem subjected to novel max-min fuzzy relational equalities as a generalization of the vertex cover problem

TL;DR

This paper introduces a new approach to optimize linear objectives under fuzzy relational equalities, generalizing the vertex cover problem, with conditions and an algorithm for finding global optima.

Contribution

It generalizes the vertex cover problem using fuzzy relation equations and provides conditions and an algorithm for global optimization.

Findings

Feasible solutions form a union of finite convex cells

Necessary and sufficient conditions for feasibility are established

An algorithm effectively finds the global optimum

Abstract

This paper considers the linear objective function optimization with respect to a novel system of fuzzy relation equations, where the fuzzy compositions are defined by the minimum t-norm. It is proved that the feasible solution set is formed as a union of the finite number of closed convex cells. Some necessary and sufficient conditions are presented to conceptualize the feasibility of the problem. Moreover, seven rules are introduced with the aim of simplifying the original problem, and then an algorithm is accordingly presented to find a global optimum. It is shown that the original problem in a special case is reduced to the well-known minimum vertex cover problem. Finally, an example is described to illustrate the proposed algorithm.