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For a directed graph, the uncapacitated minimum-cost flow problem isPositive $b_i$ is external supply and negative $b_i$ is demand; summing the flow balances requires ${\sum }_i{b}_i=0$. There are no finite upper arc capacities. Let $B$ be the oriented incidence matrix with tail $+1$ and head $-1$. For free dual potentials ${\pi }_i$, the Lagrangian isIts infimum over nonnegative flows is ${\pi }^{T}b$ exactly when every network reduced cost $r_{ij}=c_{ij}-{\pi }_i+{\pi }_j$ is nonnegative; otherwise the infimum is minus infinity. Thus the dual maximizes ${\pi }^{T}b$ subject to these inequalities. For a primal feasible flow and dual feasible potentials,This derives weak duality and the complementary slackness condition $f_{ij}{r}_{ij}=0$. A feasible pair satisfying it has equal objectives and certifies optimality.

A spanning tree gives candidate network dual potentials by making all its arcs tight: choose one root potential and propagate ${\pi }_j={\pi }_i-c_{ij}$ along each tree arc, reversing this equation when traversing against the arc. An arbitrary tree does not automatically make the non-tree reduced costs nonnegative; those inequalities must be checked. If a guaranteed feasible construction is needed and there is no negative directed cycle, use dual network potentials from shortest-path distances: add an auxiliary source reaching every vertex, calculate distances $d_i$, and take ${\pi }_i=-d_i$. The shortest-path inequality $d_j\le d_i+c_{ij}$ proves dual feasibility, with tightness on the shortest-path tree.

For the printed dashed tree, set all non-tree flows to zero and solve the balances from the leaves. This givesThe remaining arcs $46,67,52,28,89$ have zero flow. The total initial cost is $20$. This is a basic feasible solution, though degenerate because the tree flow on $68$ is zero.

Normalize ${\pi }_1=0$. Tree tightness givesThe non-tree reduced costs areThe negative reduced cost on $89$ identifies an improving entering arc for the network simplex algorithm. Adding it to the tree creates a unique undirected cycle. Sending $\theta$ along $8\to 9$ increases the flows on $15,56,68,89$ by $\theta$ and decreases those on $14,43,37,79$ by $\theta$; all other flows are unchanged. This balanced displacement preserves every node equation. Nonnegativity limits $\theta$ to the minimum of the four decreasing flows, namely $2$. Choose arc $14$ to leave among the tied limiting arcs.

After this one pivot the nonzero flows areand every other arc flow is zero. The objective decreases by $\theta r_{89}=-2$, giving cost $18$.

For the new tree, tight potentials normalized at vertex 1 areAll tree reduced costs are zero. For the non-tree arcs $14,46,67,52,28$ they are $1,1,1,2,1$, respectively, so the potentials are dual feasible. Their dual objective ismatching the primal cost. Complementary slackness therefore provesThis is the stopping test of the network simplex method: maintain a feasible tree flow, compute tight potentials, enter a negative-reduced-cost non-tree arc, make a maximal feasible cycle adjustment and exchange a limiting tree arc, until all reduced costs are nonnegative. Here exactly one improving step suffices.