Design of Minimum Cost Maximum Flow Using Floyd Algorithm

This article discusses the design of minimum cost maximum flow using Floyd's algorithm, where the labeling algorithm is employed for maximum flow computation. Floyd's algorithm serves as an efficient graph algorithm widely applied in network flow problems and other graph theory applications. The minimum cost maximum flow problem involves finding the maximum flow from a source node to a sink node in a graph while minimizing the total cost of edges traversed by the flow. The labeling algorithm provides a standard solution approach by iteratively adjusting node labels to discover augmenting paths, thereby incrementally increasing flow while reducing costs.

The implementation typically involves maintaining two key data structures: a distance matrix for Floyd's all-pairs shortest path computation and node labels for capacity tracking. The algorithm proceeds through multiple phases: first computing shortest paths using Floyd's O(n³) dynamic programming approach, then applying labeling techniques to identify feasible flow augmentation paths. Each iteration updates residual capacities and cost adjustments until optimality conditions are satisfied.

In summary, this paper presents a comprehensive methodology combining Floyd's algorithm and labeling techniques for solving minimum cost maximum flow problems. The discussion includes fundamental concepts of both algorithms and their practical application scenarios. This approach can be implemented using adjacency matrices for graph representation, with key functions handling path cost calculation and flow augmentation. The article aims to enhance readers' understanding of network flow problems and graph theory applications, providing practical value for real-world implementations.