[英]Showing NP, NP-Completeness, or NP-Hardness
Is my understanding of the three categories correct? 我对这三个类别的理解是否正确?
To show a problem X is NP: 为了表明问题X是NP:
To show a problem X is NP-Complete: 要显示问题,X是NP-Complete:
To show a problem X is NP-Hard: 为了表明问题X是NP-Hard:
You almost got it. 你几乎得到了它。
Given a problem X
, to show it is NPC, you don't need to show X ≤p L
, for some NPC problem L
. 给定问题X
,为了显示它是NPC,对于某些NPC问题L
,您不需要显示X ≤p L
。
In fact, this is guaranteed, since you already showed that X
is in NP (in 1), and you know L
is NP-Complete. 事实上,这是有保证的,因为你已经证明X
是NP(1),你知道L
是NP-Complete。 By definition of NP-Complete, this means there is a polynomial time reduction from ALL problems in NP to L
, including from X
, so basically your step (3) in proving NPC is redundant. 根据NP-Complete的定义,这意味着从NP到L
所有问题都有一个多项式时间减少,包括来自X
,因此基本上证明NPC的步骤(3)是多余的。
A more elegant way to show the statements of what needs to be done to prove each property: 一种更优雅的方式来显示证明每个属性需要做什么的陈述:
To show X
is NP: 要显示X
是NP:
To show X
is NP-Hard: 要显示X
是NP-Hard:
OR 要么
L
in NP, L ≤p X (this is done only once actually, for SAT, and is the definition of NP-Hard). 显示对于NP中的任何问题L
,L≤pX(对于SAT,这实际上只执行一次,并且是NP-Hard的定义)。 To show a problem X is NP-Complete: 要显示问题,X是NP-Complete:
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