Chapter 882: The first issue of Exploring Mathematics
The millennium problem of P=NP? conjecture is the core 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 and 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 transformation, optimization theory, statistics, etc. to achieve intelligent decision-making.
For example, graph isomorphism theory, although it still faces many challenges at present, has many applications in image processing, video analysis, finding optimal paths, network flow algorithms, efficient obstacle avoidance and other fields.
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, when she heard Xu Chuan asking about research and manuscript papers on graph isomorphism, Liu Jiaxin smiled 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 prepared it in advance.
"Let me see."
Seeing the manuscript paper, Xu Chuan couldn't wait to reach out and take it.
Indeed, compared with 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 fields.
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, a picture is a two-dimensional plane, and there are "countless" points on a picture.
If we say, assuming 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 needed. (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 is only suitable for graphs with very few nodes.
Judging from the manuscript paper in hand, when Liu Jiaxin was studying this problem, he did not bring all graph isomorphism problems into P=NP problems.
She chose to use quasi-polynomials and mapping functions to cut isomorphic modules and 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 their points 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 graphical 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 the category of P-type problems or NP-type problems.
But it is undeniable that this is a major breakthrough in the problem of graph isomorphism.
.......
After reading the manuscript paper in his hand, Xu Chuan breathed a long 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 problem of graph isomorphism for decades, and it has never happened before.
Someone has done this before.”
Liu Jiaxin shook her head gently and said: "But there is still a long way to go before we can solve this problem."
Xu Chuan smiled and said: "The road is walked step by step. If you can find a better and easier road, it means you are already on the right road."
After a slight pause, he raised the manuscript paper in his hand and asked with a smile: "This is probably not the original manuscript, can I keep it?"
This chapter is not finished yet, please click on the next page to continue reading the exciting content! "There are still many things in it that are worthy of in-depth study. I can take a good look at it during the Chinese New Year."
This manuscript paper is obviously not an original manuscript, but a complete research after sorting it out. There is no problem even if it is written as a paper and sent out.
It is impossible to fully understand all the essence of such a wonderful paper in just ten minutes.
Especially when it comes to the Millennium Problem such as the P=NP? conjecture, even if it is only a tentative proof of a secondary problem, it is of great value and deserves in-depth exploration.
Liu Jiaxin nodded and said softly: "Of course, this is brought here specially for you."
Xu Chuan smiled and said, "Then I'll accept it."
"in addition....."
As he spoke, he spread out the manuscript paper in his hand and continued: "I have some ideas about tools for 'mapping isomorphism'."
Hearing this, Liu Jiaxin moved her body, sat over, and looked at the manuscript paper in her hand close to him.
Xu Chuan pointed at somewhere on the manuscript paper and said, "For example, here, when you map subgraph isomorphisms, you use injective functions and bijective functions."
"But the function f mapped in this way will usually only satisfy the mapping relationship of the homomorphic graph, and you will need to further process it later."
"And if we use the Weyl group mapping method of algebraic varieties, we can first simplify it to..."
In the office, Xu Chuan talked a lot about the problem of graph isomorphism and the ‘quasi-polynomial graph mapping tool’ he had in hand.
In the fields of P=NP? conjecture and algorithmic mathematics, his research is indeed inferior to Liu Jiaxin.
However, as a scholar who stands at the top of the mathematics community, his way of looking at problems and comprehensiveness, as well as the feasibility of solving this problem, are not comparable to her.
Especially since Liu Jiaxin has initially perfected a simplified mathematical tool for graph isomorphism problems, it is not difficult for him to find certain points that can optimize this tool.
...
The afternoon almost passed by as the two of them discussed each other.
After talking about the last discussion point, Xu Chuan stretched out, 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 map and compare it and narrow its scope bit by bit, it may be difficult to solve this problem."
"Just like the Riemann Hypothesis, it will probably never be possible to continuously compress its critical point and push it to the 1/2 line. It requires a new method."
"So for the problem of graph isomorphism, my suggestion is that maybe you need to break out of the traditional way and find a new path."
"This quasi-polynomial graphical 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 carefully."
Xu Chuan smiled and said: "Don't rush, just take your time. If you encounter any problems during the research process, you can come to me at any time."
"Perhaps I may not be able to provide you with any help, but having one more person and one more idea will definitely make you better in mathematics."
Liu Jiaxin nodded and joked: "Of course, it would be too pitiful for me to leave such a good mathematician alone and fight alone."
Xu Chuan smiled, his eyes fell on the manuscript paper in his hand, and suddenly he 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 with."
"What's wrong?"
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"."
"It will be the first issue of Exploration Mathematics, if you will!"
Hearing this, Liu Jiaxin nodded without any hesitation and said: "No problem."
After finishing her words, she hesitated for a moment, then thought about it and asked: "Is this paper enough?"
Liu Jiaxin is very clear about the papers published in the first issue of "Exploration" and "Exploration Physics". The former is the unified theory of strong electricity, and the latter is a theory related to room temperature superconducting materials.
In contrast, her paper seems a bit lacking.
Xu Chuan smiled and shook his head and said: "If P=NP? If the conjecture research is not qualified, what else is qualified?"
"You can be more confident that it is enough to be published in any journal in the world."