A bit-capacity scaling algorithm for the constrained minimal cost network flow problem

• Published in: Network biology Vol. 11; no. 2; pp. 82 - 96
• Main Author: Tlas, Muhammad
• Format: Journal Article
• Language: English
• Published: Hong Kong International Academy of Ecology and Environmental Sciences (IAEES) 01.06.2021; International Academy of Ecology and Environmental Sciences
• Subjects: Algorithms; assignment problem; augmenting path method; Biology; bit-scaling algorithm; Flow; Graph theory; Labeling; Lower bounds; minimal cost flow problem; polynomial time algorithm; Polynomials; Shortest-path problems; transportation problem
• ISSN: 2220-8879; 2220-8879

A polynomial time algorithm for solving the minimum-cost network flow problem has been proposed in this paper. This algorithm is mainly based on the binary representation of capacities; it solves the minimum-cost flow problem in directed graph of n nodes and m directed arcs as a sequence of O(n2) shortest path problems on residual networks. The algorithm runs in O(n2mr) time, where r is the smallest integer greater than or equal to Log2B, and B is the largest arc capacity of the network. A generalization of this proposed algorithm has been also performed in order to solve the minimum-cost flow problem in a directed network subject to non-negative lower bound on the flow vector. A formulation of both the transportation and the assignment problems, as a minimal cost network flow problem has been also performed. A numerical example has been inserted to illustrate the use of the proposed method.