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Decide whether you think the following statements are true or false.Let [mathjaxinline]G[/mathjaxinline] be an arbitrary connected, undirected graph with a distinct cost on every edge. Suppose [mathjaxinline]e[/mathjaxinline] is the cheapest edge in [mathjaxinline]G[/mathjaxinline]. Then, there is a minimum spanning tree of [mathjaxinline]G[/mathjaxinline] that contains the edge [mathjaxinline]e[/mathjaxinline].

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Given a MAX 3-SAT formula $\Phi$ with k clauses, we show that the expected number of clauses satisfied by a random assignment is 7k/8. Consider a random assignment for formula $\Phi \wedge \bar{\Phi}$, the expected number of clauses satisfied is _____.

A. 7k/16
B. 7k/8
C. 7k/4
D. 7k/2

There are 8 statements and we have no knowledge about them. We randomly guess them true or false. So the possibility that we correctly guess 6 statement is _____.

A. 1/256
B. 9/256
C. 37/256
D. 93/256

Given the same 3 men and 3 women as exercise 1. Which of the following misrepresentation can make Grace obtain better partner for herself in the returned matching?

A. Grace lies "I prefer David to Jack"
B. Grace lies "I prefer Jack to Bob"
C. Grace lies "I prefer David to Bob"

Gale-Shapley algorithm finds______.

A. a perfect matching without stability
B. possible different assignments among all executions
C. the man-optimal assignment which is a stable matching

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