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.

https://pastebin.com/hNYrCjhL

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.

Of course, because I am a selfish person, I'd say I appreciate most his work on convex body chasing (see "Competitively chasing convex bodies" on the Wikipedia link), because it follows up on some of my work.

Objectively, you should check his conference submission record, it will be a huge number of A*/A CORE rank conferences, which means the best possible in TCS. Or the prizes section on Wikipedia.

> Last time I had to fill out a government form in Canada (...)

Without any evidence, let me argue why maybe it shouldn't. In the past, a common opinion that I have heard is that open source is more secure because all the code is out in the open.

The recent xzutils backdoor attempt [1] kind of led me to believe it's not really true, it's only true if many good-actor eyeballs, which are willing to donate their time for public benefit, are on the code.

Almost all of the government's code that I interact with are web apps that are potential targets of foreign adversaries -- tax filing web apps, prescription + vaccination scheduling web apps, family benefit applications, and more. (This is not in Czechia, but close.)

Now, would I want to read that web app code? Not at all, I couldn't care less about it. However, foreign adversaries would love to immediately start analyzing it. Extracting the entire country's health data or tax data would be a goldmine.

And even though there probably are several people actively paid to maintain security of these systems, I feel that the foreign adversarial agents would be much more motivated (and better paid) than government employees/software developers.

You could make a opt-out for national-security purposes for the code, but I feel almost all the code a government works on would have such an impact when compromised.

[1]: https://en.wikipedia.org/wiki/XZ_Utils_backdoor

(Disclaimer: I am a huge supporter of open source in general, contributed to the Linux ecosystem in the past and in my current job as an academic, almost everything I do is available out in the open in some way or another.)

I have the same trajectory as you -- multiple countries in the EU, working in academia -- but different experiences for sure. Or at least a mixed bag.

Let me list them in order of how much English sufficed:

1. The Netherlands -- common knowledge is that their English is top notch and anecdotally it was the case as well, I also got by purely with English.

2. Germany -- their English is also good but I needed German in edge cases. One edge case was finding an apartment (not speaking German simply pushed you down the list of candidates, even with a full time job in academia). Another one were university rules and announcements; not every email was in English, but arguably easy to get by with modern translation tools.

3. Czechia & Poland -- English is good among the professors but the percentage of locals at the university level is so high that most internal meetings, announcements, local seminars take place in the local language. In my experience, non-faculty university staff (department secretaries, payroll, entrance security) usually strongly dislike speaking English or outright do not speak it at all.

---

I've omitted some more cases where local languages are required. If you live in a country, you will eventually interact with the healthcare sector, where the language experience will likely mimic the experience at the workplace (for the countries above, it would be in the same order for the healthcare sector).

Another case is government bureaucracy. For most of the EU countries I've been to, the official language of the country is their local language and only their local language. This means that government employees are not required to speak any other language other than the official one to you, plus you might be required to fill in forms and communicate in the official language if you want to talk to them.

In my experience, the helpful/good ones may try to communicate with you in English but if you need something from them or if the bureaucrat had a bad day, you better start talking in the official language.

After that episode, where I tried myself to get rid of them, I am much more convinced that Youtube is fine with all but the worst scammers, and don't buy any of the "they're just low on manpower" arguments anymore.

[1]: https://news.ycombinator.com/item?id=34943047

> Hi! I'm a PhD student studying computer science at Rice University.

This means that we are on the same career path (I am currently an assistant professor in theoretical CS in Europe). I wish you of course best of luck!

Here is the harshest truth about teaching I learned during my PhD:

If you are focusing on teaching too much, you are setting yourself up for failure.

This sounds cruel, and in fact I am much like you, I love teaching and I love self-improvement and it is quite easy for me to invest time into my teaching prep, presentation, and more and see measurable results in class quality and usually also student feedback.

However, at least in my neck of the woods (i.e. Europe), almost all gates and gatekeepers for you as a PhD student, and later postdoc, are checking your research. At some places they really do expect you to have K publications in the top 3 CS conferences or you will not be considered at all -- and it seems these thresholds are only getting higher. Here I mean for example invitation-only workshops, postdoc positions with top advisors, and later also permanent positions.

On the other hand, if you are a talented scientist, they usually only care that your teaching skills are at the bare minimum -- have you taught something? Yes? Great.

Now orator/presentation skills are critical and presenting a coherent lecture plan might be useful for a final presentation at an interview for a permanent position. But even there, it is more about you knowing what you want to teach and how it complements the department than about your past achievements (i.e., how much you have put in a course previously).

My PhD advisor usually said that he likes to dig into teaching when research is not going well. I agree with that -- teaching really is fulfilling to me and I love to improve my class and see people happy with it, and research is all about global ranking (which is tough on anyone's psyche) and generating progress which is the fun part but sometimes takes a long time. However, at your stage of your career, the research really can't go slow.

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PS: If the author reads this, since it is a self-post, your class sounds really nice and it is actually one I would have loved to attend. My research is in online algorithms -- a field which you can rephrase as seeing some theoretical problems as two player games between a solver and an adversary -- and among other things I would like to consider utilizing all the techniques of chess solvers (which cannot evaluate the game fully, but "almost") and transfer it to other areas of online algorithms.

Matchmaking decreases the odds you meet a cheater for low rank players, and significantly increases it for higher rank players -- and since there's fewer of them due to the Bell curve, they are going to feel the cheaters that much more.

If you just rely on rank and not on anti-cheat efforts, you'd be just destroying one of the loyal cores of the playerbase, one which is also quite vocal online.

From my personal experience of thousands of hours in competitive FPS shooters on PC, there is no point in ranking where playing against a cheater becomes fair or fun.