What is a polynomial-time reduction? (NP-Hard + NP-complete)

Here we introduce a "polynomial-time reduction," which is one in which takes polynomial time (obviously). We also introduce the notion of NP-hardness and NP-completeness. We then show that if a problem is NP-complete, and is also in P, then P = NP.

▶ABOUT ME◀
I am a professor of Computer Science, and am passionate about CS theory. I have taught many courses at several different universities, including several sections of undergraduate and graduate theory-level classes.

[EasyTheory]

Thanks to my supporters Yuri (Youtube) and Bruno, Timmy, Micah (Patreon) for making this video possible! If you want to contribute, links are in the video description.

[mariajosepav]

Hi, I have a final on Tuesday, and I just wanted to say thank you. I got more out of this than a 2 hour lecture. I like how you get to the point in a way that is easy to understand and I am very thankful that I found your channel.

[MrPlanes]

Right in time for my final in a couple days, thank you!

[pauminguetmarti]

Extremely helpful. Very concise and well explained. Best video i've found on the topic. Thanks for making it.

[Kleinnnn]

Thank you for the clarity! As I'm studying this, it would be helpful to have some examples of actual problems being NP-hard but not complete, and also some examples of languages to which all NP problems can be reduced, but that are not in NP, together with the class to which they belong. Btw, very nice contents, keep it up!

[YuriValentines]

This video explains just what i needed for my final in complexity and computability in two days. Thank you so much, king, it really was helpful!!!

[robblake]

Thank you very much! I was given the definitions of NP-hard and NP-complete with no context or real explanation, so they never really made sense to me. This video explained very well.

[serhiis.]

The best video I've ever seen about the theory of algorithms

[mprerr]

You da goat bossman, ty for all of these videos

[manuelsteele]

I am a PhD student in data science at Arizona State. Is that your Alma Mater too? The data science program is new and began in 2021. I was admitted with the first class and have six master's degrees. I am currently enrolled in CSE 550 which is on combinatorial theory with heavy focus on NP topics. I am using this channel to supplement the lectures. Thanks for the great teaching!

[nicholascunningham]

"It's NP but it's not in P" ok, I get what you're saying, but the wording of that is so confusing 😅 Could say something like "It's in the set NP but not in the subset P"

[shiewhun]

I got lost precisely at 2:19. What is Sigma Star? And then what is Gamma Star? I am not claiming you're not a great explainer. You are, and it's why I'm subscribed. I am just saying at 2:19, I stopped being able to follow this video.