Honest Solver Scope - Integer Programming & Branch-and-Bound

Branch and bound is still exact where the tree is drawable; the solver's contribution is executing the same method at a scale no one draws by hand. The chapter's claims stop at one pinned instance with a stated three-leg certificate.

highlighted = computed this step

Branch and bound is still exact

Nothing here replaces the earlier chapters: branch and bound is still exact on the pinned 2-variable instances, where the relaxation picture and the tree are drawable. Why: the method is not wrong at five variables; the pictures are what stop scaling.

\text{exact where drawable}

What the solver adds

At five variables with these domains the integer lattice alone holds 1625829408 points, and the branch-and-bound tree is far too large to draw by hand; a real solver's branch and bound is what makes the instance tractable. Why: the tree is still the method — just executed by a solver with pruning and bounds instead of drawn node by node.

\text{same method, solver-scale execution}

Transfer boundary

The honest boundary: this chapter certifies one pinned five-variable instance with a three-leg certificate — own re-verification, the solver's bound meeting its objective, and this repo's own independent general branch-and-bound reaching the same optimal value. It says nothing about solver runtimes or 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 number shown is exact.

\text{exact here; no runtime promise elsewhere}