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I am trying to prove the NP-Completeness of problem [A].

I know there is a well-known NP-Complete problem called problem [B].

I can model [A] into an instance of [B] ([B] is a very general problem and when I assign some edges a weight value then it becomes [A]).

Is this sufficient to say [A] is NP-Complete?

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  • $\begingroup$ Try emulating what you saw in class. $\endgroup$ – Yuval Filmus Dec 25 '18 at 6:05
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To show $A$ is NP-complete, you need to prove the following:

  1. $A \in \mathbf{NP}$
  2. $A$ is NP-hard: any problem $B \in \mathbf{NP}$ is poly-time (many-one) reducible to $A$

    or

    equivalently (because reduction relations are transitive): there is an NP-hard problem $B$ which is poly-time (many-one) reducible to $B$.

You have shown neither, so you cannot say $A$ is NP-complete.

What you did do is a many-one reduction from $A$ to $B$, but you want the other way around. In addition, you also need to show the reduction is efficient, that is, takes polynomial time; you did not explicitly state whether this is the case. And, of course, you must not forget to show $A \in \mathbf{NP}$ (which is usually the easier part).

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