Speed up simplex via pricing rules and leaner phase-1#2
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Two algorithmic improvements to the exact fraction-free simplex, both preserving the exact optimal answers on every test instance. Pricing (solver/tableau.ml, bin/simplex.ml): - Add --rule dantzig (most-positive reduced cost) and --rule steepest (steepest-edge approximation, maximising c^2 / ||col||^2 via exact integer cross-multiplication). Both use a Bland index tie-break for anti-cycling. Bland stays the default. On the 100x100 instance: bland 3034 pivots/4.7s -> dantzig 478/1.3s -> steepest 237/0.37s. Phase-1 restructure (solver/tableau.ml): - Build negative-RHS constraints already in artificial-basic form (negate the row so the artificial has coefficient +1 and RHS >= 0) and repair the phase-1 objective in a single construction pass. This removes the forced setup pivots that previously brought each artificial into the basis one by one (~21% of all pivots; large steepest 237 -> 186). Final answers verified identical across all 61 test problems. The pricing/phase-1 changes alter pivot sequences (and counts) on problems with artificial variables, so the affected golden .out files are regenerated; only pivot counts change, the solutions and objective values are unchanged. Investigated but deliberately not pursued (measured, documented in commit history): GCD content removal is a no-op (Edmonds/Bareiss already leaves no removable common content, 1-3 bits out of ~430), and sparse CSC storage loses at the ~45-50% mid-solve density of these instances.
This was referenced Jul 2, 2026
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Two algorithmic improvements to the exact fraction-free simplex, both preserving the exact optimal answers on every test instance.
Pricing (solver/tableau.ml, bin/simplex.ml):
Bland stays the default. On the 100x100 instance: bland 3034 pivots/4.7s ->
dantzig 478/1.3s -> steepest 237/0.37s.
Phase-1 restructure (solver/tableau.ml):
The pricing/phase-1 changes alter pivot sequences (and counts) on problems with artificial variables, so the affected golden .out files are regenerated; only pivot counts change, the solutions and objective values are unchanged.
Investigated but deliberately not pursued (measured, documented in commit history): GCD content removal is a no-op (Edmonds/Bareiss already leaves no removable common content, 1-3 bits out of ~430), and sparse CSC storage loses at the ~45-50% mid-solve density of these instances.