K-Coloring is the property that a graph can be colored with k colors such that no two neighboring nodes get the same color. The K-Coloring problem is a decision problem, ie a yes/no question.
The chromatic number of a graph is the lowest k for which it has a k-coloring.
Clearly, if you have an algorithm for one, you have an algorithm for the other.
The question was: is it faster to compute k-coloring than to compute its lowest (actual) k, ie its chromatic number.
Forests, trees, and bipartite graphs are 2-colorable. Planar graphs are 4-colorable. It is NP-complete to check if the chromatic number of a planar graph is 3.
There's a very interesting open problem, Hadwiger's conjecture that essentially says that the chromatic number is the clique (minor) number (whatever that means).
At the end of brev paper there is a section on the LLM usage.
Now, this paper will undergo peer review. And thus, when published, we will trust the result as well as any other published result in mathematics. However, while humans are not infallible, LLMs have a tendency to spit out confident stuff which looks correct. Thus I worry when it is used this way to produce proofs. Peer review is not perfect, and may not be tuned to catch LLM’s style of errors.
There is a solution to this, which is to formalise the result and get it machine verified. And with LLMs, I daresay this is going to be best practice moving forward.
> Peer review is not perfect, and may not be tuned to catch LLM’s style of errors
This summarizes I think a lot of the challenges with validating LLM output. We hear “humans make mistakes too”, but I would agree with you that our human detection of human-made mistakes and LLM-made mistakes is unlikely to have the same coverage.
The real problem is that human mistakes generally occur more frequently given the difficulty of the task, whereas LLM mistakes are somewhat random, like the carwash problem, because LLMs cannot truly reason.
I'd rather have a human on my team for whom I can reasonably surmise what tasks they're good at than have a robot who randomly gets shit wrong.
I worded a questions similar to this with a person in the finance industry. He was very receptive of AI since it is one of the easiest ways to grow stock portfolio.
Say you have two assists. One that reads the reports and one that uses AI to summarize the content. You are sitting in a high value meeting. You only can have one assistant with you in the meeting. Which one would you pick, the persons that read the reports or the person who only used AI?
Not much faster.
Any k-coloring algorithm of complexity F(n) can be used to create a chromatic number algorithm of complexity lg(N)F(N) simply by bisecting on N.
For those not familiar:
K-Coloring is the property that a graph can be colored with k colors such that no two neighboring nodes get the same color. The K-Coloring problem is a decision problem, ie a yes/no question.
The chromatic number of a graph is the lowest k for which it has a k-coloring.
Clearly, if you have an algorithm for one, you have an algorithm for the other.
The question was: is it faster to compute k-coloring than to compute its lowest (actual) k, ie its chromatic number.
Forests, trees, and bipartite graphs are 2-colorable. Planar graphs are 4-colorable. It is NP-complete to check if the chromatic number of a planar graph is 3.
There's a very interesting open problem, Hadwiger's conjecture that essentially says that the chromatic number is the clique (minor) number (whatever that means).
> Forests, trees, and bipartite graphs are 2-colorable.
More precise: a graph is bipartite if and only if it is 2-colorable (this can actually be used as a definition).
Since forests are bipartite, and trees are forests, the other two statements follow.
At the end of brev paper there is a section on the LLM usage.
Now, this paper will undergo peer review. And thus, when published, we will trust the result as well as any other published result in mathematics. However, while humans are not infallible, LLMs have a tendency to spit out confident stuff which looks correct. Thus I worry when it is used this way to produce proofs. Peer review is not perfect, and may not be tuned to catch LLM’s style of errors.
There is a solution to this, which is to formalise the result and get it machine verified. And with LLMs, I daresay this is going to be best practice moving forward.
> Peer review is not perfect, and may not be tuned to catch LLM’s style of errors
This summarizes I think a lot of the challenges with validating LLM output. We hear “humans make mistakes too”, but I would agree with you that our human detection of human-made mistakes and LLM-made mistakes is unlikely to have the same coverage.
The real problem is that human mistakes generally occur more frequently given the difficulty of the task, whereas LLM mistakes are somewhat random, like the carwash problem, because LLMs cannot truly reason.
I'd rather have a human on my team for whom I can reasonably surmise what tasks they're good at than have a robot who randomly gets shit wrong.
I worded a questions similar to this with a person in the finance industry. He was very receptive of AI since it is one of the easiest ways to grow stock portfolio.
Say you have two assists. One that reads the reports and one that uses AI to summarize the content. You are sitting in a high value meeting. You only can have one assistant with you in the meeting. Which one would you pick, the persons that read the reports or the person who only used AI?
Not much faster. Any k-coloring algorithm of complexity F(n) can be used to create a chromatic number algorithm of complexity lg(N)F(N) simply by bisecting on N.