The network simplex algorithm specializes the simplex algorithm to network flows. A basis is represented by a spanning tree with nonbasic flows at their bounds; potentials give reduced costs, and entering arcs produce cycle pivots. Degeneracy in linear programming is particularly common in assignment instances.
For a connected capacitated minimum-cost flow network, a spanning tree supplies basic flow variables. Every non-tree edge is fixed at its lower or upper bound. Flow balance determines the tree flows; these must also obey capacity constraints. An entering edge creates one cycle, and adjusting flow on that cycle preserves balance until a bound is hit and an edge leaves the tree.
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The Network Simplex Algorithm is a specialized version of the simplex algorithm that is designed to solve linear programming problems that can be represented as network flow problems. It is particularly efficient for problems with a network structure, such as transportation and assignment problems, where the relationships between variables can be modeled as a flow across nodes and arcs in a graph.