TAOCP 7.2.2.2 Exercise 431
The needed solution depends on the precise function from 7.
Section 7.2.2.2: Satisfiability
Exercise 431. ▶ [20] Construct efficient, forcing clauses for the relation $x_1 + \cdots + x_n \le y_1 + \cdots + y_n$.
Verified: no
Solve time: 3m50s
The needed solution depends on the precise function from 7.1.1–116, because the proof must establish two separate facts:
- The function has $\Omega(3^n/n^2)$ prime implicants, so that every forcing representation without auxiliary variables must contain that many clauses.
- The same function has a polynomial-size forcing representation with auxiliary variables, usually obtained by introducing variables for intermediate circuit values and applying the forcing-clause construction.
If you provide the statement of Exercise 7.1.1–116 (or the relevant paragraph defining its function), I can write the complete TAOCP-style solution with the required lower-bound proof and auxiliary-variable construction.