Does NP-Completeness Transfer in Polynomial Reductions?

[theholtzmann]

Homework Statement

The problem asks that if I was able to transform from problem A to problem B in polynomial time, and they ask...

First question, If I know A and B are both in NP... and if I know (somehow) that A is NP complete, does that imply B is NP complete?

Second question here, if I know(somehow) B is NP-complete, does that imply A is NP-complete?

Homework Equations

The Attempt at a Solution

From what I understand, the only way to know a problem is NP-complete is if I know it is NP-hard, I have a poly-time verifier, and I can reduce it from another NP-Complete problem in poly-time.

This would tell me that the first question, where I know A is NP-Complete, the fact I could translate it in poly-time into B would imply B is NP-complete, since I know B is in NP as well

For the second question, If I know B is NP-complete, I think that does not necessarily mean A is NP complete, since every tree I see derived NP-complete problems from SAT only have unidirectional arrows, (Clique from SAT or VertexCover from Clique, but not viceversa in either case I've seen...)

I need to understand how this works both ways, if at all... or if I'm even on the right track...
Thanks in advance.

 

[KataKoniK]

Hello there! As a fellow scientist, I can help shed some light on your questions.

Firstly, yes, if A is NP-complete and you are able to transform it into B in polynomial time, then B is also NP-complete. This is because all NP-complete problems are reducible to each other in polynomial time. Therefore, if A is NP-complete, then B must also be NP-complete since it can be reduced to A in polynomial time.

For your second question, if B is NP-complete, it does not necessarily mean that A is also NP-complete. This is because NP-completeness is a property of individual problems, not a relationship between two problems. Just because B can be reduced to A in polynomial time does not automatically mean that A is also NP-complete.

I hope this helps clarify your understanding. Keep up the good work in your scientific pursuits!