题目内容
If we have a magic machine that can solve 3-SAT problem in unit time. Which of the following problem can NOT be solve if we only use this machine polynomial times? (Assume P ≠ NP)
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Suppose we are given an instance of the [mathjaxinline]s-t[/mathjaxinline] shortest path problem on a directed graph [mathjaxinline]G[/mathjaxinline]. We assume that all edge cost are positive and distinct. Let [mathjaxinline]P[/mathjaxinline] be a minimum-cost [mathjaxinline]s-t[/mathjaxinline] path for this instance. Now suppose we replace each edge cost [mathjaxinline]c_e [/mathjaxinline] by its square, [mathjaxinline]c_e^2 [/mathjaxinline], thereby creating a new instance of the problem with the same graph but different costs.Decide whether you think the following statement is true or false.[mathjaxinline]P[/mathjaxinline] must still be a minimum-cost [mathjaxinline]s-t[/mathjaxinline] path for this new instance.[mathjaxinline]P[/mathjaxinline] must still be a minimum-cost [mathjaxinline]s-t[/mathjaxinline] path for this new instance.
If we want to proof problem $Y$ can be solved in polynomial-time via a reduction from problem $X$, which of the following statement is NOT necessary?
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