Solved at Scale

Honest Solver Scope

Flow-stream is a different specialized tool from the general CP-SAT solver: this problem class has its own dedicated exact algorithm. The chapter's claims stop at one pinned instance, with no performance promises beyond it.

highlighted = computed this step

A different tool

Flow-stream wraps a dedicated polynomial-time exact max-flow algorithm — a different tool from the general CP-SAT solver used in the other capstone chapters, not better or worse: this problem class has its own exact algorithm. Why: choosing the tool that matches the problem class is itself part of honest modeling.

the right exact tool for the class\text{the right exact tool for the class}
The decoded flowsOne row per arc: capacity and the literal decoded flow.one certified flow, total 60capacityflowN1 to N22020N1 to N33030N1 to N41010N2 to N3400N2 to N53020N3 to N41010N3 to N52020N4 to N350N4 to N52020

The book's method still applies

Nothing here replaces the earlier chapters: Ford-Fulkerson and the min-cut reading are still exact — they produced this chapter's certificate legs live at build time. Why: the solver's role at this scale is to do on real data what the book's algorithm does on the teaching network, with the same theorem behind both.

same theorem, both runs\text{same theorem, both runs}
The decoded flowsOne row per arc: capacity and the literal decoded flow.one certified flow, total 60capacityflowN1 to N22020N1 to N33030N1 to N41010N2 to N3400N2 to N53020N3 to N41010N3 to N52020N4 to N350N4 to N52020

Transfer boundary

The honest boundary: this chapter certifies one pinned 5-node instance — one decoded flow, re-verified, matched by the book's own run and its own cut — and says nothing about performance on larger networks. Why: a certified flow on a small instance is exact knowledge about that instance, not a performance promise about a different one. Pixel positions are rounded for layout; every number shown is exact.

exact here; no performance promise elsewhere\text{exact here; no performance promise elsewhere}
The decoded flowsOne row per arc: capacity and the literal decoded flow.one certified flow, total 60capacityflowN1 to N22020N1 to N33030N1 to N41010N2 to N3400N2 to N53020N3 to N41010N3 to N52020N4 to N350N4 to N52020