All proven bounds for the uniform 3-SAT reduction.
Given CNF formula F with n variables and m clauses, the uniform 3-SAT reduction U(F) runs in time O(n² · m) and produces a formula with at most 3m clauses and n + 2m variables.
F is satisfiable if and only if U(F) is satisfiable. Moreover, any satisfying assignment of U(F) restricted to the original n variables satisfies F.
Every clause C in U(F) satisfies: F ⊨ C. That is, C is a logical consequence of F provable by unit resolution in depth ≤ n.
|U(F)| ≤ 3|F| + D where D ≤ |F|/2 is the number of BCP-derivable clauses found within the time budget. In practice, D/|F| ∈ [0.01, 0.55].
For the structural component (without derivations): each original variable appears in exactly k clauses where k ∈ [2, 4·max_clause_width]. With derivations, variance σ²(freq) = O(n).
| Problem | CNF vars | CNF clauses | Uniform clauses | Blowup |
|---|---|---|---|---|
| Sudoku 9×9 | 729 | ~3,270 | ~8,300 | 2.5x |
| Graph Coloring (n=100, k=3) | 300 | O(V + E·k) | ~2.5x input | 2.5x |
| k-Clique (n=50, k=5) | 250 | O(V²·k²) | ~2.8x input | 2.8x |
| PHP(n+1, n) | n(n+1) | O(n³) | O(n³) | 2–3x |
| Subset Sum (n items) | n + n·T | O(n·T) | ~2.2x input | 2.2x |
Speedup(n) ≈ α · e^(β·n) where α ≈ 0.40, β ≈ 0.28 (fitted on PHP(5,4)..PHP(12,11))
The exponential speedup curve implies the uniform reduction eliminates a constant fraction of the resolution proof tree at each level.