TAOCP 7.2.2.2 Exercise 520
In the direct encoding, each variable $x_i$ is represented by Boolean variables indicating its possible values.
Section 7.2.2.2: Satisfiability
Exercise 520. ▶ [24] Every instance of SAT corresponds in a natural way to an integer programming feasibility problem: To find, if possible, integers $x_1, \ldots, x_n$ that satisfy the linear inequalities $0 \le x_j \le 1$ for $1 \le j \le n$ and
$$l_1 + l_2 + \cdots + l_k \ge 1 \qquad \text{for each clause } C = (l_1 \vee l_2 \vee \cdots \vee l_k).$$
For example, the inequality that corresponds to the clause $(x_1 \vee \bar{x}_3 \vee \bar{x}_4 \vee x_7)$ is $x_1 + (1-x_3) + (1-x_4) + x_7 \ge 1$; i.e., $x_1 - x_3 - x_4 + x_7 \ge -1$.
Sophisticated "IP solvers" have been developed by numerous researchers for solving general systems of integer linear inequalities, based on techniques of "cutting planes" in high-dimensional geometry. Thus we can solve any satisfiability problem by using such general-purpose software, as an alternative to trying a SAT solver.
Study the performance of the best available IP solvers, with respect to the 100 sets of clauses in Table 6, and compare it to the performance of Algorithm C in Table 7.
Verified: no
Solve time: 3m43s
Solution
In the direct encoding, each variable $x_i$ is represented by Boolean variables indicating its possible values. Write
$$ x_{i,j}=1\iff x_i=j, $$
where $0\le i<p$ and $1\le j\le d$ if the domain has size $d$. The direct encoding already contains the clauses forcing each $x_i$ to have exactly one value. The all-different constraint adds the requirement that two different variables cannot choose the same value.
For two distinct variables $x_i$ and $x_k$, and for any value $j$, the assignment
$$ x_{i,j}=x_{k,j}=1 $$
must be forbidden. The corresponding clause is
$$ \bar{x}{i,j}\vee \bar{x}{k,j}. $$
Taking all pairs of variables and all possible values gives the complete encoding:
$$ \bigwedge_{0\le i<k<p}\ \bigwedge_{1\le j\le d} (\bar{x}{i,j}\vee\bar{x}{k,j}). $$
These clauses are sufficient. Suppose the direct variables represent an assignment satisfying the clauses. If two variables $x_i$ and $x_k$ had the same value $j$, then the direct encoding would require
$$ x_{i,j}=1,\qquad x_{k,j}=1. $$
The clause
$$ \bar{x}{i,j}\vee\bar{x}{k,j} $$
would then be false, contradicting satisfiability. Hence no two variables receive the same value.
They are also necessary. Given any assignment satisfying the all-different constraint, no pair of variables $x_i,x_k$ has the same value. Therefore, for every $j$, at least one of $x_{i,j}$ and $x_{k,j}$ is false, so every clause
$$ \bar{x}{i,j}\vee\bar{x}{k,j} $$
is satisfied.
Thus the all-different constraint in the direct encoding is enforced by adding one binary clause for every pair of variables and every possible value:
$$ \boxed{\bar{x}{i,j}\vee\bar{x}{k,j}\qquad (0\le i<k<p,\ 1\le j\le d).} $$
This completes the proof. ∎