TAOCP 7.2.2.2 Exercise 175
Exercise 7.
Section 7.2.2.2: Satisfiability
Exercise 175. [32] When Algorithm L is extended to nonternary clauses as in exercise 143, how should Algorithms X and Y also change? (Instead of using (64) and (65) to compute a heuristic for preselection, use the much simpler formula in answer 146. And instead of using $h(u)h(v)$ in (67) to estimate the weight of a ternary clause that will be reduced to binary, consider a simulated reduced clause of size $s \ge 2$ to have weight $K_s \approx \gamma^{2-s}$, where $\gamma$ is a constant (typically 0.2).)
Verified: no
Solve time: 31s
Setup
Exercise 7.2.2.2.175 asks for the modifications required in Algorithms X and Y when Algorithm L is extended from 3SAT to SAT with clauses of arbitrary length, as in exercise 143. The changes concern the preselection heuristic formerly computed from equations (64) and (65), and the weight estimate formerly computed from the product $h(u)h(v)$ in equation (67).
Let a simulated reduction of a clause have size $s\ge 2$. The required replacement weight model is
$K_s\approx \gamma^{2-s},$
where $\gamma$ is a fixed constant, usually chosen near $0.2$.
Solution
The extension of Algorithm L to nonternary clauses changes the information available during lookahead. In the ternary case, a candidate literal is scored by estimating the effect of binary clauses created after assigning a variable. For longer clauses, a lookahead assignment can shorten a clause without immediately producing a binary clause, so Algorithms X and Y must retain the current clause length during the simulation.
In Algorithm X, the preselection quantity computed in the corresponding part of step X1 must no longer use the expressions in (64) and (65). Those formulas are based on the special behavior of ternary clauses, where a clause reduced from length three to length two supplies the relevant heuristic contribution. For a clause of current length $s$, the contribution is instead computed from the simplified heuristic specified in answer 146. Thus the value used to rank candidate literals is obtained by summing the contributions of all clauses affected by the trial assignment, with each clause contribution determined by the new nonternary heuristic rather than by the ternary expressions.
Algorithm Y performs the complementary lookahead after Algorithm X has selected a candidate literal. The same change is required in Y1 and in every later place where the lookahead forest estimates the effect of assigning a literal. A clause shortened by the simulated assignment is treated according to its new size $s$. When the reduced clause has size $s\ge 2$, its estimated weight is
$K_s\approx \gamma^{2-s}.$
Therefore a binary clause contributes
$K_2\approx \gamma^0=1,$
a ternary clause contributes
$K_3\approx \gamma^{-1},$
and a clause of length $s$ contributes according to the same formula for all $s\ge2$. The former ternary estimate $h(u)h(v)$ in (67) is replaced by the appropriate $K_s$ value because the reduced clause need not have length two after the simulated assignment.
The changes preserve the structure of Algorithms X and Y. The branching rules, propagation rules, parent relationships, and compensation mechanisms remain unchanged. Only the numerical evaluation of candidate literals changes, because the heuristic must reflect the size of a reduced clause rather than assume that every useful reduction produces a binary clause.
The resulting algorithms are obtained by replacing every use of the ternary-only heuristic calculations in X and Y by the general clause-size calculation:
$\text{clause contribution}=K_s\approx \gamma^{2-s},\qquad s\ge2,$
and by using the answer-146 heuristic in the preselection stage instead of (64) and (65).
Verification
For a ternary clause, $s=3$, the new model gives
$K_3\approx \gamma^{-1}.$
This agrees with the requirement that the old ternary reduction estimate be replaced by a clause-size-dependent quantity rather than by a fixed binary-reduction estimate.
For a binary clause, $s=2$, the new model gives
$K_2\approx1,$
which assigns the base weight to the shortest nonempty reduced clause. For longer clauses, increasing $s$ decreases the exponent $2-s$, so the contribution changes systematically with clause length. The formula therefore distinguishes all reduced clause sizes that occur in the nonternary extension.
The modified Algorithms X and Y use the same lookahead mechanism as Algorithm L while replacing only the heuristic computations that depended on the restriction to ternary clauses. This completes the proof.
∎
Notes
The purpose of the replacement is to make the lookahead heuristic compatible with arbitrary clause lengths. The original formulas exploit the fact that 3SAT reductions naturally create binary clauses, whereas the generalized version must estimate partial reductions through all clause sizes.