Honest Solver Scope - Bin Packing: First-Fit Decreasing, Step by Step

FFD also reaches the optimum on this instance, but that is favorable, not guaranteed. The certificate does the proving, and the chapter's claims stop at one pinned instance with no runtime or generality promises.

highlighted = computed this step

FFD reaches it here

FFD from the earlier chapters also reaches 4 bins on this instance, with a different arrangement than the solver's: the two packings group the items differently. Why: this instance has more than one optimal packing, and both are certified by the same bin-count proof; landing on an optimum here is a favorable result on this instance, not a general guarantee.

\text{FFD bins}=4

Not a guarantee

FFD is a heuristic in general. This packing is optimal only because the exact lower-bound certificate matches the 4 bins used here. Why: on other instances FFD can use more bins than the optimum, so the certificate, not the heuristic, carries the proof.

\text{heuristic}+\text{certificate}=\text{proof}

Transfer boundary

The honest boundary: this chapter certifies one pinned 11-item instance via the exhaustive search, the lower-bound certificate, and the solver's own bound, and says nothing about solver runtimes or about larger instances. 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 item size, fill, total size, and bound is exact.

\text{exact here; no runtime promise elsewhere}