TAOCP 7.2.2.2 Exercise 482

In the direct encoding, each variable $x_i$ is represented by Boolean variables indicating its possible values.

taocpmathematicsalgorithmsvolume-4hard

Section 7.2.2.2: Satisfiability

Exercise 482. ▶ [26] [26] (Erdős discrepancy patterns.) The binary sequence $y_1 \ldots y_t$ is called strongly balanced if we have $|\sum_{j=1}^{k}(2y_j - 1)| \le 2$ for $1 \le k \le t$.

  • a) Show that this balance condition needs to be checked only for odd $k \ge 3$.
  • b) Describe clauses that efficiently characterize a strongly balanced sequence.
  • c) Construct clauses that are satisfied by $x_1 x_2 \ldots x_n$ if and only if $x_d x_{2d} \ldots x_{\lfloor n/d \rfloor d}$ is strongly balanced for all $d$.

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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. ∎