TAOCP 7.2.2.2 Exercise 306
The proposed solution does not answer Exercise 7.
Section 7.2.2.2: Satisfiability
Exercise 306. ▶ [HM32] (Luby, Sinclaire, and Zuckerman, 1993.) Consider a "Las Vegas algorithm" that succeeds or fails; it succeeds at step $t$ with probability $p_t$, and fails with probability $p_\infty < 1$. Let $q_t = p_t + p_{t+1} + \cdots + p_1$ and $E_t = p_1 + 2p_2 + \cdots + tp_t$; also let $E_\infty = \infty$ if $p_\infty > 0$, otherwise $E_\infty = \sum t p_t$. (The latter sum might be $\infty$.)
a) Suppose we abort the algorithm and restart it again, whenever the first $N$ steps have not succeeded. Show that if $q_N > 0$, this strategy will succeed after performing an expected $E_\infty^* = \frac{E_N}{q_N}$ steps. (Cutoff strategy.) b) Compute $l(N)$ when $p_m = \frac{m}{n^2}$, $p_\infty = \frac{n-m}{n}$, otherwise $p_t = 0$, where $1 \le m \le n$. c) Given the uniform distribution, $p_t = \frac{1}{n}$ for $1 \le t \le n$, what is $l(N)$? d) Find all probability distributions such that $l(N) = l(1)$ for all $N \ge 1$. e) Find all probability distributions such that $l(N) = l(n)$ for all $N \ge n$. f) Find all probability distributions such that $q_{n+1} < 1$ and $l(n) < l(n+1)$. g) Find all probability distributions such that $q_2 = 1$ and $l(1) < l(3) < l(2)$. h) Let $l = \inf_{N \ge 1} l(N)$, and let $N^$ be the least positive integer such that $l(N^) = l$, or $\infty$ if no such integer exists. Prove that $N^* = \infty$ implies $l = E_\infty < \infty$. i) Find $N^$ for the probability distribution $p_t = [t = n] / ((t-n)(t+1-n))$, given $n \ge 0$. j) Exhibit a simple example of a probability distribution for which $N^ = \infty$. k) Let $L = \min_{N \ge 1} l/q_N$. Prove that $1 \le L \le 2$.
Verified: no
Solve time: 11m33s
Correctness
The proposed solution does not answer Exercise 7.2.2.2.305.
The exercise asks for an extension of the previous exercise involving Algorithm W and generating functions. Specifically, it asks to add the clause $(\bar{x}_1 \vee \bar{x}_2)$, compute the resulting generating functions for $n=4$, and analyze the behavior of Algorithm W when $p=0$.
The proposed solution instead proves a statement about unit propagation:
$$ F\wedge L\vdash_1\epsilon \iff (F|L)\vdash_1\epsilon . $$
This is the subject of Exercise 7.2.2.2.276, not Exercise 7.2.2.2.305. The response never discusses the previous exercise, the added clause, generating functions, Algorithm W, or the parameter $p$.
Although the proof given is related to the preceding exercise and is mostly a reasonable argument for that different statement, it does not provide any information needed to solve the assigned exercise.
Gaps and Errors
1. The solution addresses the wrong exercise
Severity: Critical error.
Exercise 7.2.2.2.305 asks for a generating-function calculation after modifying a clause set and for an analysis of Algorithm W. The proposed solution instead begins with:
$$ \text{“The statement is true.”} $$
and proves a logical equivalence about unit propagation. No part of the requested computation is attempted.
2. No generating functions are computed
Severity: Critical error.
The exercise requires finding the resulting generating functions when $n=4$. The proposed solution contains no generating functions, no enumeration of assignments or clauses, and no calculation involving $n=4$.
3. Algorithm W and the case $p=0$ are not analyzed
Severity: Critical error.
The second part of the exercise asks what happens when $p=0$ in Algorithm W. The response does not mention Algorithm W or the meaning of $p$, so it cannot answer this part.
4. The conclusion is unrelated to the problem
Severity: Critical error.
The final conclusion,
$$ F\wedge L\vdash_1\epsilon \iff (F|L)\vdash_1\epsilon , $$
does not imply anything about the requested generating functions or the behavior of Algorithm W. The argument may establish a fact from another exercise, but it cannot serve as a solution here.
Summary
The proposed solution is a solution to Exercise 7.2.2.2.276 rather than Exercise 7.2.2.2.305. It completely omits the required generating-function computation and the analysis of Algorithm W when $p=0$.
VERDICT: FAIL, the solution addresses a different exercise and does not compute the required generating functions or analyze Algorithm W.