NotOscarWilde
Joined 13 years ago
Joined 13 years ago
A Polynomial-Time Algorithm for Three-Machine Unit-Job Scheduling [1]
Since some people talk about small numbers that pop up in integer multiplication results, here a completely different number appears:
Theorem 1.1. Let an explicitly listed finite directed acyclic graph specify the precedence constraints on n >= 1 nonpreemptive unit-length jobs on three identical machines. There is a uniform deterministic algorithm that constructs a feasible schedule of minimum makespan. Given also an integer deadline 1 <= T <= n, it decides feasibility exactly and returns a schedule whenever the answer is affirmative. Both tasks can be performed in O((L + 2)^150020) steps on a deterministic multitape Turing machine, where L is the total binary input length.
That is some crazy exponent -- plus an interestingly old computational model to boot; not something that is natural to most of us. I have no capacity to check its correctness today, but I hope it is true purely for the exponent.
[1]: https://github.com/openai/math/blob/main/preprints/A-polynom...
The situation right now is pretty crazy, for example this established TCS professor [1] has at least 4 arXiv submissions co-authored by him in September [2]. This includes the recent breakthrough on the matroid secretary problem, which has a pretty interesting AI story of its own, if you haven't seen it yet [3].
Of course prolific professors can work around this by working with younger researchers who upload the work, but this just makes the rate limit a solo author bottleneck, which feels a bit weird.
[1]: https://en.wikipedia.org/wiki/Mohammad_Hajiaghayi
[2]: https://arxiv.org/search/cs?query=Hajiaghayi&searchtype=auth...
[3]: Concurrent Discovery Disclosure: The proof of the main result in this manuscript was obtained in a conversation with ChatGPT-6 Astra on Tuesday, September 15, 2026 at 1:02 AM PDT. We then prepared this manuscript for public release, with the intent of uploading it on the morning of Thursday, September 17, 2026. In the early morning hours of September 17, while finalizing the submission, we discovered the manuscript of Abdi, Banihashem, Hajiaghayi, and Mittal, uploaded on September 16, 2026, which contains the same result via an essentially identical approach. We are sharing our manuscript nonetheless in case our exposition is of independent utility to the community, and we hope this experience stimulates broader discussion about concurrent discovery in the AI era. (From https://arxiv.org/abs/2609.20797 .)
I had a son recently -- everything indeed went fine, but my wife and I did read up on both the risks and the permanent changes to women's bodies that having a child brings. Among other things, a C-section is not recommended an unlimited number of times, and it is classified as a surgical procedure for a reason.
You can pay somebody to get a biological child that you cannot have, but at the same time you are paying them to carry some risk for you within their bodies and possibly undergo surgery for you. I am not anti-surrogacy myself, but it just is not that simple as a win-win.
But it is a moot point anyway. Cook and Levin are very well known already in TCS, and credit is not directly enumerable like money, so "more than a lot of credit" doesn't make too much sense.
For this problem in particular, asking the right kind of question was really important for the field and led to a lot of discoveries even before it will be answered.
I think about this on the regular -- I know the answer is currently "you own the code, so you have to understand it", but to unlock the true productivity multiplier, in the future, the answer has to be "no one really".
I think about it using the concepts from my job (academia) -- to actually have PhD student-level intelligence means that you have to trust that it does a good enough job that you can focus on other stuff. Professors often bring the correct ideas or intuitions, but they have to trust the PhD student to write the code and/or fill in the gaps in the proofs -- they can advise them on the high-level issues during a consultation, but that's about it.
I am pretty bad at working in the current LLM workflow -- it is tough for me to focus on reading a TCS paper for review, keeping all the details and invariants in my head, but every 5-10 minutes go to my PC, completely switch contexts/projects, read the code and think about the LLM's comments, suggest the next step, and then go back to reading.
We are wading into philosophy here, but I believe this analogy doesn't track in this case -- my suspicion from this blog post and others is that already today, the Pro level thinking models are a positive multiplier to your research output similar to how the models one level lower are a multiplier to one's programming output.
Maybe one can someday use the cheaper models similar to how you can use cheaper models than Opus/5.5 and still be nearly as productive as a programmer -- but I am trying and failing doing exactly that for research questions.
Thank you.
An aside: It was a very nice gesture and completely unexpected by me, so even if it doesn't work out, it made my day. I personally believe that kind gestures have a lot of power.
Back on topic: There is a real danger of the gap between rich and poor universities significantly widening in all fields if the rich can afford Pro level models, or even hardware that can run their own comparable models, and this being fiscally inaccessible to the rest.
One can sweep this under the rug by blaming the educational funding but this just shoots down all discussion. Even if GDP of a country goes up by a lot -- such as Poland -- it takes time before any budget benefit trickles to the education budget, and with some governments it might never do.
I believe Microsoft et al do have the most power here to boost affordable access to AI for researchers on a large scale; the fact that they cut some too expensive models (Opus, 5.5) from their academic benefits package is a grim omen. I do realize they would like universities to pay them also, and ultimately the universities should do that -- but then we are back at the institutional level of the problem.
You seem to have a good estimate in your head; I definitely do not.
From personal experience, ChatGPT 5.5 (the Plus tier) is excellent for programming tasks and also for various teaching related tasks but I have not observed the research benefits that Tim Gowers has when I asked it questions in my area of expertise. So the costs are definitely higher than a few dozen $ a month per PhD/professor.
You might be right that universities should immediately spring into action and demand funding for research level AI resources and hardware. One thing you might be mistaken in is that public universities are unfortunately very inflexible institutions; one reason for this is that they have a large internal leadership structure AND they are funded by the state, so even if the entire university agrees on something, the funding is at the whim of the ministry of education and thus the current political leadership.
I probably will erase the contents in a few days.
Even if you just drop an email and it doesn't work out, I appreciate this gesture so much. Thank you.
There are several other philosophical arguments against what you propose but I do not wish to go down that route.
Paying for Pro from any of my current academic budgets is completely ouf of the field of reality here -- all budgets tend to have restricted uses and software payments fit into very few categories. Effectively, I'd have to ask for a brand new grant and hope the grant rules allow for large software payments and I won't encounter an anti-AI reviewer; such a thing would take one year at least.
As a nail to the coffin, I was "denied" all Claude Opus recently as part of Microsoft's clampdown on individual (and academic) use of Copilot.
(Chagpt 5.5 Plus does not seem sufficient for any deeper investigations into new research topics, I've tried.)
Apologies for the rant.
Fully agree, in fact, this has literally happened to me a week ago -- ChatGPT was confidently incorrect about its simple lock structure for my multithreaded C++ program, and wrote paragraphs upon paragraphs about how it works, until I pressed it twice about a (real) possibility of some operations deadlocking, and then it folded.
> Every time a major announcement comes out saying so-and-so model is now a triple Ph.D programming triathlon winner, I try using it. Every time it’s the same - super fast code generation, until suddenly staggering hallucinations.
As an university assistant professor trying to keep up with AI while doing research/teaching as before, this also happens to me and I am dismayed by that. I am certain there are models out there that can solve IMO and generate research-grade papers, but the ones I can get easy access to as a customer routinely mess up stuff, including:
* Adding extra simplifications to a given combinatorial optimization problem, so that its dynamic programming approach works.
* Claiming some inequality is true but upon reflection it derived A >= B from A <= C and C <= B.
(This is all ChatGPT 5, thinking mode.)
You could fairly counterclaim that I need to get more funding (tough) or invest much more of my time and energy to get access to models closer to what Terrence Tao and other top people trying to apply AI in CS theory are currently using. But at least the models cheap enough for me to get access as a private person are not on par with what the same companies claim to achieve.