Generalized Path Optimization Problem for a Weighted Digraph over an Additively Idempotent Semiring

Abstract

In this paper, a generalized path optimization problem for a weighted digraph (i.e., directed graph) over an additively idempotent semiring was considered. First, the conditions for power convergence of a matrix over an additively idempotent semiring were investigated. Then we proved that the path optimization problem is associated with powers of the adjacency matrix of the weighted digraph. The classical matrix power method for the shortest path problem on the min-plus algebra was generalized to the generalized path optimization problem. The proposed generalized path optimization model encompasses different path optimization problems, including the longest path problem, the shortest path problem, the maximum reliability path problem, and the maximum capacity path problem. Finally, for the four special cases, we illustrate the pictorial representations of the graphs with example data and the proposed method.