Solved with Flow-Stream - Max-Flow, Min-Cut

An external tool wrapping a dedicated polynomial-time exact max-flow algorithm solves the pinned network; with no search parameters at all, the run is inherently deterministic. The flows shown are decoded from the run's own output.

highlighted = computed this step

Invocation

The invocation is simply flow_stream_solve maxflow.json — no parameters at all. Why: this tool wraps a dedicated polynomial-time exact max-flow algorithm, not a general search, so the run is inherently deterministic and there is no seed or worker count to fix.

\text{dedicated exact algorithm, no params}

Solver result

The solver reports status OPTIMAL with total flow 60. Why: OPTIMAL here is the exact algorithm's guarantee by construction. Unlike the CP-SAT tools used elsewhere in this track, this binary's output carries no timing or search-statistics field to exclude at all — status, total flow, and per-arc flows are the entire response, and every one of them is deterministic.

\text{status OPTIMAL, total flow }60

The decoded flows

Reading the flow column top to bottom: 20, 30, 10, then 0 on N2 to N3, and 20, 10, 20, then 0 on N4 to N3, and 20. Why: these are the literal decoded per-arc flows from the real solve, and two arcs honestly carry zero — an arc in the network need not carry flow in every optimal answer.

\text{decoded flows, exactly as solved}

Diagram note

The table adds a flow column next to each arc's capacity; the row labels are the same endpoint labels as the previous lesson, and the caption's total is recomputed from the decoded flows on every build. Pixel positions are rounded for layout; every number shown is exact.

\text{per-arc flows, exact}