But it's really fascinating.
It appears to be relatively good at problems where finding the initial answer is difficult, but verifying whether a candidate answer is correct is easy. In particular, AI feels very strong in matching-type problems, almost like fuzz testing. As seen in Terence Tao's conversations, it has a massive advantage in rapidly substituting and testing various models.
Given these strengths, I feel it would be highly effective for problems like the Hadamard matrix of order 668, the Lonely Runner conjecture, and the Graceful Tree conjecture.
Perhaps the unsolved problems I mentioned will be cracked in the near future? It is fascinating.
Solved? https://epoch.ai/frontiermath/open-problems/hadamard
gwt4life•23m ago
zygentoma•12m ago
> The problem’s definition is simple: There are k servers located at points of a metric space. At each time step, a request arrives at a point of the metric space. An online algorithm must serve the request immediately by moving a server to the requested location, without knowledge of future requests. The goal is to minimize the total distance traveled by servers.
So metric space is anything where you can measure a distance, so you know the distances between all servers and the distance from the request to all servers. Could be direct distance, could be travel time …
Easiest to just imagine just some (eg. n=5) servers on a plane. A request pops up somewhere on the plane. Which server do you move there, such that the total distance moved by servers is as low as possible in the end after a sequence of requests.
fxwin•8m ago
> The [k-server] problem’s definition is simple: There are k servers located at points of a metric space. At each time step, a request arrives at a point of the metric space. An online algorithm must serve the request immediately by moving a server to the requested location, without knowledge of future requests. The goal is to minimize the total distance traveled by servers.
> The k-server conjecture states that a deterministic online algorithm can achieve competitive ratio k on every metric space.
I only had to look up what "competitive" means in this context, and wikipedia [0] had this to say about it:
> An algorithm is competitive if its competitive ratio—the ratio between its performance and the offline algorithm's performance—is bounded.
The ratio by which this performance is bounded for a k-competitive algorithm is k (plus some constant) [1]. We can consider the analogy of k support technicians ("servers) located in different locations (in metric space): The conjecture/theorem states that in any metric space (Not necessarily two- or three-dimensional), there exists an online algorithm that results in travelled distances of no more than roughly k times that of the optimal distance if all requests were known in advance.
[0] https://en.wikipedia.org/wiki/Competitive_analysis_(online_a...
[1] https://www14.in.tum.de/personen/albers/papers/brics.pdf Section 1.1