Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 38 / 3 / b

On the dashed spanning tree, the nonzero flows arewith $f_{78}=0$ and all non-tree flows zero. The tree balances give supply four at vertex $1$, demands one at $5$ and $8$, and demand two at $6$. Its cost is $25$.

Choose ${\pi }_1=0$ and make the tree network reduced costs zero. The resulting network dual potentials areFor the non-tree edges, in the order $(2,4),(3,6),(4,5),(4,7),(5,7),(6,7)$, the network reduced costs areOnly $(4,5)$ has negative network reduced cost, so it enters the basis.

Adding this edge to the tree creates the graph cycle $4\to 5\to 2\to 1\to 3\to 4$. Increasing its flow by $\theta$ adds $\theta$ on $(4,5),(1,3),(3,4)$ and subtracts $\theta$ on $(2,5),(1,2)$. These signed changes preserve every flow balance. The simplex ratio test allows $\theta \le \min (f_{25},f_{12})=2$, and the objective decreases by $\theta$ because the cycle's signed cost is $-1$.

Take $\theta =2$ and remove $(2,5)$ from the basis. The other tied edge $(1,2)$ remains a zero-flow basic edge; this is a legitimate degenerate tree basis and requires no extra pivot. The new flow isIts cost is $8+4+6+4+1=23$. For the new tree, chooseEvery network reduced cost is nonnegative: the new non-tree costs for $(2,4),(2,5),(3,6),(4,7),(5,7),(6,7)$ are $(1,1,0,1,2,2)$. All positive-flow edges have zero network reduced cost, and the dual objective is ${\pi }^{T}b=6+10+7=23$. Thus complementary slackness gives