TAOCP 7.2.2.2 Exercise 484
In the direct encoding, each variable $x_i$ is represented by Boolean variables indicating its possible values.
Section 7.2.2.2: Satisfiability
Exercise 484. [22] [22] (Graph quenching.) A graph with vertices $(v_1, \ldots, v_n)$ is called "quenchable" if either (i) $n = 1$; or (ii) there's a $k$ such that $v_k - v_{k+1}$ and the graph on $(v_1, \ldots, v_{k-1}, v_{k+1}, \ldots, v_n)$ can be quenched; or (iii) there's an $l$ such that $v_l - v_{l+3}$ and the graph on $(v_1, \ldots, v_{l-1}, v_{l+2}, v_{l+3}, \ldots, v_n)$ can be quenched.
- a) Find a graph on four vertices that is quenchable although it is $3$-chromatic.
- b) Construct clauses that are satisfiable if and only if a given graph is quenchable. Hint: Use the following three kinds of variables for this model-checking problem:
$x_{t,i,j} = [v_i \to v_j \text{ at time } t]$, for $1 \le i < j \le n-t$; $q_{t,k} = [\text{a quenching move of type (ii) leads to time } t+1]$; $s_{t,l} = [\text{a quenching move of type (iii) leads to time } t+1]$.
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. ∎