Great Country Academician

P=NP? Conjecturing this millennium problem is at the heart of the field of computer mathematics.

The entire problem can be divided into N sub-difficulties, of which four are the most famous.

They are the factorization problem of large positive integers, the graph isomorphism problem, the discrete logarithm problem, and the Manhattan network problem.

Among them, the large number decomposition problem and the graph isomorphism problem are the most famous, because the solutions to these two problems involve many basic issues in cryptography and complexity theory.

Solving them can greatly promote the development of computer science, algorithms and other aspects.

Not to mention other things, just like intelligent driving, environmental perception and decision-making algorithms are the key to autonomous vehicles achieving intelligent driving.

The environment perception algorithm uses on-board sensors and cameras to obtain information such as roads, obstacles, traffic signs, etc., and performs perception and modeling.

The decision-making algorithm makes driving path planning, communication coordination, avoidance decisions and other operations based on the results of environmental perception.

These algorithms usually involve multiple technologies such as multi-sensor fusion, state estimation, and machine learning.

It is also necessary to use a large number of mathematical models and algorithms such as matrix operations, image transformations, optimization theory, and statistics to achieve intelligent decision-making.

For example, graph isomorphism theory, although it still faces many challenges, has many applications in image processing, video analysis, finding optimal paths, network flow algorithms, and efficient obstacle avoidance.

Breakthroughs in mathematical tools have brought more advanced and efficient calculation methods to the algorithmic judgment of AI intelligent driving.

This is also the reason why the intelligent driving of Chuanhai Network Technology Company can quickly catch up with other companies that have already made in-depth deployment in just two or three years.

In the office, Liu Jiaxin smiled when she heard Xu Chuan asking about research and manuscript papers on graph isomorphism, and took out the sorted manuscript papers from the backpack she carried with her.

She knew that Xu Chuan's attention would definitely be focused on this, so she was prepared in advance.

"Let me see."

Seeing the manuscript paper, Xu Chuan couldn't wait to reach out and take it.

Indeed, compared to breakthroughs in the field of intelligent driving, he is more focused on the problem of intention isomorphism.

This involves the answer to a millennium problem and also involves more application areas.

Smart driving is just a part of its applications.

Looking through the manuscript paper in his hand, Xu Chuan had an interested look in his eyes.

".Given two graphs G =(V G, E G) G=(V_G, E_G)G=(V

G,E G) and H =(V H,E H) H=(V_H,E_H)H=(V H,E H). "

"If there is a mapping from G to H: V G→ V H, satisfying: (v i)= v i′, (v j)= v′j"

"It's a bit interesting. Instead of taking a broader approach to P-type problems, we use quasi-polynomials and mapping functions to cut isomorphic modules."

"This method is somewhat similar to the way of studying the weak Riemann hypothesis?"

Looking at the manuscript paper in his hand, Xu Chuan muttered to himself.

The graph isomorphism problem, in simple terms, is to give two graphs and ask whether they are exactly the same.

And how to check whether two given graphs are isomorphic and exactly the same?

One of the best ways is to simply compare each point to match all possible corresponding nodes in another graph.

But as we all know, pictures are two-dimensional planes, and there are ‘countless’ points on a picture.

If we assume a graph with N nodes, according to this matching calculation method, the number of matches is the factorial of N (1*2*3** N), which is far more than the order of magnitude of N.

Assuming there are only 10 nodes in the graph, more than 3.6 million possible matching checks are already required. (1*2*3*10)

And if a graph has 100 nodes, the number of possible matches will far exceed the number of atoms in the visible universe.

Therefore, this brute force method is very impractical and only suitable for graphs with very few nodes.

Judging from the manuscript paper in hand, when Liu Jiaxin was studying this problem, she did not bring all graph isomorphism problems into P=NP problems.

She chose to use quasi-polynomials and mapping functions to cut isomorphic modules into images while treating these ‘contrast points’ as blocks of ‘images’.

Then simulate the four-color theorem, starting from some small nodes in the first picture, and "painting" each of them with a different color.

Then assume that there are one-to-one corresponding points in the second graph, start looking for isomorphisms in it, and mark these corresponding nodes with the same color after finding them.

The algorithm iterates until all possible guesses are finally verified.

This is a more efficient algorithm than previous graph isomorphism problems, and the key lies in a mathematical tool in these manuscripts.

"Quasipolynomial graph mapping method."

This tool completes efficient algorithms for graph isomorphism by connecting polynomials and mapping tools.

Although it does not solve the problem of graph isomorphism, it does not even completely classify this problem into a P-type problem or an NP-type problem.

But it is undeniable that this is a major breakthrough in the problem of graph isomorphism.

After reading the manuscript in his hand, Xu Chuan breathed a sigh of relief, looked at Liu Jiaxin with a smile on his face, and said: "I didn't expect you to give me another surprise. The mathematical community has been studying the graph isomorphism problem for decades, and no one has ever achieved this level."

Liu Jiaxin shook her head gently and said: "But there is still a long way to go to solve this problem."

Xu Chuan smiled and said: "The road is taken little by little. Being able to find a better and simpler road means that you are on the right road."

After a slight pause, he raised the manuscript in his hand and asked with a smile: "This should not be the original manuscript, can I keep it?"

"There are still many things in it that are worth further study. I can take a good look at it during the Chinese New Year."

This manuscript is obviously not the original manuscript, but a complete study after sorting it out. It is no problem to even write it into a paper and send it out.

And it is impossible to fully comprehend all the essence of such a wonderful paper in just a dozen minutes.

Especially when it comes to the millennium problem such as the P=NP? conjecture, even if it is just an experimental proof of a secondary problem, it is of great value and worth further exploration.

Liu Jiaxin nodded and said softly: "Of course, this is specially brought to you."

Xu Chuan smiled and said: "Then I will accept it."

"In addition."

As he said, he spread out the manuscript in his hand and continued: "Regarding the tools for 'mapping isomorphism', I have some ideas here."

Hearing this, Liu Jiaxin moved her body, sat over, and looked at the manuscript in his hand close to him.

Xu Chuan pointed to a place on the manuscript paper and said, "For example, here, when you are mapping the subgraph isomorphism, you use the method of injective function and bijective function."

"But the function f mapped in this way usually only satisfies the mapping relationship of the isomorphic graph, and you need to further process it later."

"If the Weyl group mapping method of algebraic cluster is used, it can be simplified to"

In the office, Xu Chuan talked a lot about the graph isomorphism problem and the "quasi-polynomial graph mapping method tool" in his hand.

In the field of P=NP? conjecture and algorithmic mathematics, his research is indeed not as good as Liu Jiaxin.

However, as a scholar standing at the top of the mathematics world, his way of looking at problems and comprehensiveness, as well as the feasibility of solving this problem, are not comparable to hers.

Especially when Liu Jiaxin has preliminarily perfected a simplified mathematical tool for the graph isomorphism problem, it is not difficult for him to find some points that can optimize this tool.

The afternoon passed by in the mutual discussion between the two.

After finishing the last discussion point, Xu Chuan stretched himself, picked up the already cold tea from the coffee table, took a sip to moisten his throat and said.

"I feel that the graph isomorphism problem may be similar to the Riemann hypothesis to some extent. If we simply compare and map it and narrow its scope little by little, it may be difficult to solve this problem."

"Just like the Riemann hypothesis, it is probably impossible to constantly compress its critical point and push it to the 1/2 line. It requires a new way."

"So for the graph isomorphism problem, my suggestion is that maybe you need to jump out of the traditional way and find a new way."

"This quasi-polynomial graph mapping method is excellent, but it may not be enough to solve this problem."

After thinking seriously for a while, Liu Jiaxin nodded and said, "Well, I'll think about it."

Xu Chuan smiled and said, "Don't worry, just take your time. If you encounter any problems in the research process, you can come to me at any time."

"Maybe I can't necessarily provide you with any help, but one more person and one more idea will definitely be better in mathematics. "

Liu Jiaxin nodded and said jokingly: "Of course, it would be too pitiful for me to go it alone without such a great mathematician."

Xu Chuan smiled, his eyes fell on the manuscript in his hand, and suddenly remembered another thing. He looked at Liu Jiaxin and said: "By the way, there is one more thing I want to ask you for help."

"What's the matter?"

Xu Chuan smiled and said: "If it is convenient, I need you to sort out this paper, write it into a formal paper, and then submit it to "Exploration·Mathematics". "

"If you like, it will be the first issue of Exploration·Mathematics!"

Hearing this, Liu Jiaxin nodded without any hesitation and said: "No problem."

After speaking, she hesitated for a moment, thought about it and asked again: "Is this paper enough?"

Liu Jiaxin knew very well what the papers published in the first issues of "Exploration" and "Exploration·Physics" were. The former was the theory of strong electric unification, and the latter was a theory related to room temperature superconducting materials.

In contrast, her paper seemed a little insufficient.

Xu Chuan smiled and shook his head, saying, "If the P=NP? conjecture research is not qualified, what else is qualified?"

"You can be more confident, it is enough to be published in any journal in the world."