Solved at Scale

Honest Solver Scope

Brute force is still exact where it is cheap to run; the solver's contribution is reaching the same answer at a scale where brute force alone eventually stops being checkable. The chapter's claims stop at one pinned instance with a stated three-leg certificate.

highlighted = computed this step

Brute force is still exact

Nothing here replaces the earlier chapters: exhaustive enumeration is still exact on the pinned 4-site instance, where every subset was checked by hand-legible arithmetic. Why: the method is not wrong at ten alternatives; it is the same all-subsets idea, just executed on more of them.

exact at every scale checked here\text{exact at every scale checked here}
Chosen alternativesThe decoded 0/1 solution, one column per alternative.one certified solution, cost 45x1x2x3x4x5x6x7x8x9x10chosen0001000101

What the solver adds

At 10 alternatives, brute force still finishes in a blink; the number of subsets doubles with every added candidate, so a real solver's value only shows up once that doubling has run long enough to make brute force no longer checkable — a threshold this chapter does not pin down. Why: this chapter does not claim the solver was NEEDED here — only that it reaches the same, independently-checkable answer.

same answer, headroom for bigger instances\text{same answer, headroom for bigger instances}
Chosen alternativesThe decoded 0/1 solution, one column per alternative.one certified solution, cost 45x1x2x3x4x5x6x7x8x9x10chosen0001000101

Transfer boundary

The honest boundary: this chapter certifies one pinned instance with a three-leg certificate — own re-verification, the solver's bound meeting its objective, and this repo's own exhaustive brute force reaching the same optimal cost. It says nothing about solver runtimes, approximation guarantees, or instances too large to brute-force. Why: an optimum 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 displayed coverage, cost, and comparison is exact.

exact here; no runtime or scale promise elsewhere\text{exact here; no runtime or scale promise elsewhere}
Chosen alternativesThe decoded 0/1 solution, one column per alternative.one certified solution, cost 45x1x2x3x4x5x6x7x8x9x10chosen0001000101