How do you explain a topic to outsiders that your own mind can barely grasp? Our editorial team faced this challenge this month. We are attempting it and hope we have succeeded.
Many thanks to our developer Max, who broadened our horizons with the topic suggestion “The P versus NP Problem”
Before we delve into this specific topic, we first want to shed light on the term “Millennium Problem.”
The term Millennium Problems currently refers to seven unsolved problems in mathematics listed in 2000 by the Clay Mathematics Institute (CMI) in Cambridge. The institute has promised a prize of one million US dollars for the solution of each of these problems, provided that they are published in a specialist journal.
The list includes these problems:
- the proof of the Birch and Swinnerton-Dyer conjecture from number theory
- the proof of the Hodge conjecture from algebraic geometry
- Analysis of the existence and regularity of solutions to the initial value problem of the three-dimensional incompressible Navier-Stokes equations
- the solution to the P-NP problem of computer science
- the proof of the Poincaré conjecture in topology
- the proof of the Riemann hypothesis of number theory
- the exploration of the Yang-Mills equations
Only one of these problems, namely the proof of the Poincaré conjecture in topology, has been solved so far. The Russian mathematician Grigori Yakovlevich Perelman was able to prove in 2002 that the conjecture is correct. After three teams successfully verified the solution, Perelman was to be awarded the promised prize money in 2010, even though he had only published the solution on the Internet. However, Perelman declined the money and the associated award.
Let’s now take a closer look at the P versus NP problem in computer science, which has been unsolved for decades. This is a so-called decision problem. The question that arises is whether the class of problems that can be solved algorithmically with relatively little effort (“polynomial time”) (the class P) is equivalent to the class of problems that, although not necessarily with little effort, can at least be checked with a reasonable effort (“non-deterministic polynomial”) (the class NP).
To illustrate the problem, one can, for example, visualise the so-called “Knapsack problems” in computer science. Imagine you are planning a hike and need to pack your backpack. The question is: Which items fit in the backpack and how much weight can you still pack? You can easily answer this question by simply packing the backpack and checking the total amount of weight. However, the real challenge is to predict which items you should choose to achieve the optimal weight. This is a typical NP problem.
The question posed by the Millennium Problem P versus NP is therefore: Can a computer solve an NP problem quickly and efficiently, or are these problems fundamentally more time-consuming, even for a powerful computer? Answering this question could revolutionise the world of computer technology.
If it turns out that P equals NP, this would be a breakthrough in solving many difficult problems, including optimising machine learning and cryptography. It would even mean that the secrets of science and technology could be unlocked like never before and our world could be improved by the computing power of computers.
On the other hand, answering the question that P is not equal to NP would mean that there is a fundamental limit to the computer world. Some problems are simply too complex for a computer to solve quickly and efficiently.
That is why the Millennium Problem P versus NP is so significant and interesting. It is a challenge that tests the limits of mathematics and computer science and helps us to expand our knowledge and improve our world through the incredible power of computers.
As we all know, a wide variety of AI systems are currently conquering the market. With regard to the problem described, in particular, AI could be the key. Perhaps an AI software will succeed in proving the assumption that P equals NP. Or one of the upcoming AI models itself could be proof that an NP problem can be solved just as quickly and efficiently by a powerful computer as a P problem.
