TAOCP 7.2.2.2 Exercise 487
In the direct encoding, each variable $x_i$ is represented by Boolean variables indicating its possible values.
Section 7.2.2.2: Satisfiability
Exercise 487. ▶ [27] [27] There are $\binom{92}{2} = 4216,368$ ways to place eight queens on a chessboard. Long ago, W. H. Rouse asked which of these leaves the maximum number of vacant squares to remain unattacked. [See W. W. Rouse Ball, Mathematical Recreations and Problems, third edition (London: Macmillan, 1896), 109–110.]
Every subset $S$ of the vertices of a graph has three boundary sets defined thus:
$$\partial S = \text{the set of all edges with exactly one endpoint} \in S;$$ $$\partial_{\text{out}} S = \text{the set of all vertices} \notin S \text{ with at least one neighbor} \in S;$$ $$\partial_{\text{in}} S = \text{the set of all vertices} \in S \text{ with at least one neighbor} \notin S.$$
Find the minimum and maximum sizes of $\partial S$, $\partial_{\text{out}} S$, and $\partial_{\text{in}} S$, over all 8-element sets $S$ in the queen graph $Q_8$ (exercise 7.1.4–241). Which set answers Turton's question?
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Solve time: 3m45s
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. ∎