@rrwilliams

Professor @MITEECS and @MIT_CSAIL. Complexity, algorithms, and related math. I'll let you know when P != NP (and when it's not)

 Cambridge, MA
csail.mit.edu/~rrw/
Joined April 2008
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  • Followers8.9K
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A number of people (grad students in particular) said they enjoyed my talk at ICALP. So maybe you'll like it too. A little unusual, it's about how/why I ended up working on circuit complexity. youtube.com/live/0DrFB2Cp7tg… (Skipping the ceremonial stuff that happened before the talk)

ICALP LICS FSCD 2024 - Award Talks

Enjoy the videos and music you love, upload original content, and share it all with friends, family, and the world on YouTube.

youtube.com
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Funny, I've been calling it the mathslopcalypse. To each their own. Can anyone honestly say they found a paper among the hundreds in the release that's *not* a huge pain to read (without AI assistance)?
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It's beyond amazing to see so many big problems solved in some form, but this feels like a DDOS attack on mathematicians. Sure, I can iterate Astra/Opus to simplify proofs and get all the notation properly defined, but why? Never had to before; we would just reject the paper.
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For what it's worth, OpenAI's one-tape TM simulation (assuming correct) immediately implies RAM time t is in SPACE[t^{4/5}] github.com/openai/math/blob/… Not sure why they didn't mention this corollary! That was the whole point of my question.

math/preprints/Simulating-One-Tape-Time-in-Two-Fifths-Power-Space-September-25-2026/article.pdf at...

Contribute to openai/math development by creating an account on GitHub.

github.com
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Yeah, page 63 of the paper should have been page 2 or 3
Gautam Kamath@thegautamkamathOct 6
A very interesting result, and a very interesting release scheme.
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The Alon-Yuster-Zwick color coding trick can be used to get 2^{O(sqrt(n))} size depth-3 AC circuits for every symmetric Boolean function!! arxiv.org/pdf/2609.34029 Just the latest exhibit that "circuit lower bounds are hard to prove because they are false"
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I've used the AYZ trick in so many papers, even tried to use it to get this circuit bound about two years ago (a simpler version of the trick lets you compute "At-Least-k" with e^k*poly(n) size depth 3 circuits), but I gave up too quickly. Wow.
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PSA: If you put your name on a paper claiming a breakthrough, you damn well better write a paper that other people can read. Otherwise, you will likely not get the kind of attention you seek. (Advice I've always given to my PhD students. Applies regardless of AI usage.)
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I agree, P/NP seems unlikely to be resolved soon. There's been essentially zero direct progress on P!=NP, so there's not much for an advanced model (or 10k parallel advanced models) to work with unless P=NP. But I could see L/NP, BPP/NEXP, P/PSPACE, etc falling not long from now
Lance Fortnow@fortnowSep 11
While P vs NP will remain out of the reach of AI, there are other complexity problems, like separating NP from L (log space) or BPP from NEXP, that might be more tractable and would still make an incredible splash.
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Which do you think is more likely? (I genuinely don't have an opinion)
79%Lean FLT and NS correct
21%Bug in the Lean kernel
620 votes • Final results
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Nice😎
Lance Fortnow@fortnowAug 9
Isik Ulusan, a U Mass student, created Complexle, like Wordle but you guess complexity classes. For each guess you get hints (set-theoretic inclusion, the type of model it is defined on, uniformity, etc.) for a total of 6 guesses. Give it a try. iulusan.github.io/complexle/…
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Wow x2!
Data Structures and Algorithms Papers@AlgorithmPapersJul 30
Breaking the 2ⁿ barrier for graph k-coloring Kevin Pratt arxiv.org/abs/2607.27159 [𝚌𝚜.𝙳𝚂]
We show that for all k, there exists εₖ > 0 such that graph k-coloring can be solved by a randomized algorithm with one-sided error in time O((2-εₖ)ⁿ). Prior to this work and independent concurrent work of Zamir [arXiv, 2026], exponential improvements over the 2ⁿ ⋅ poly(n)-time algorithm of Björklund, Husfeldt, and Koivisto [SIAM Journal on Computing, 2009] were only known for k ≤ 6.

ALT We show that for all k, there exists εₖ > 0 such that graph k-coloring can be solved by a randomized algorithm with one-sided error in time O((2-εₖ)ⁿ). Prior to this work and independent concurrent work of Zamir [arXiv, 2026], exponential improvements over the 2ⁿ ⋅ poly(n)-time algorithm of Björklund, Husfeldt, and Koivisto [SIAM Journal on Computing, 2009] were only known for k ≤ 6.

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Ryan Williams @rrwilliams.bsky.social retweeted
Data Structures and Algorithms Papers@AlgorithmPapersJul 30
Breaking the 2ⁿ barrier for graph k-coloring Kevin Pratt arxiv.org/abs/2607.27159 [𝚌𝚜.𝙳𝚂]
We show that for all k, there exists εₖ > 0 such that graph k-coloring can be solved by a randomized algorithm with one-sided error in time O((2-εₖ)ⁿ). Prior to this work and independent concurrent work of Zamir [arXiv, 2026], exponential improvements over the 2ⁿ ⋅ poly(n)-time algorithm of Björklund, Husfeldt, and Koivisto [SIAM Journal on Computing, 2009] were only known for k ≤ 6.

ALT We show that for all k, there exists εₖ > 0 such that graph k-coloring can be solved by a randomized algorithm with one-sided error in time O((2-εₖ)ⁿ). Prior to this work and independent concurrent work of Zamir [arXiv, 2026], exponential improvements over the 2ⁿ ⋅ poly(n)-time algorithm of Björklund, Husfeldt, and Koivisto [SIAM Journal on Computing, 2009] were only known for k ≤ 6.

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Wow! This was a major open problem in exponential-time algorithms. Congrats!
Or Zamir@OREAXEAXJul 29
Exciting result: k-Coloring is in (2-eps_k)^n time, where eps_k>0 for every fixed k! Algorithms faster than the 2^n time needed to compute the chromatic number were previously known only for k<=6. arxiv.org/abs/2607.25973

k-Coloring is Faster than Computing the Chromatic Number

We prove that $k$-coloring on $n$-vertex graphs has a randomized algorithm running in time $(2-\varepsilon_k)^n$, where $\varepsilon_k>0$ for every fixed $k$. Previously, only the cases $k\leq 6$...

arxiv.org
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I worry that Fable 5 may be getting too emotional to solve my problems 🧐
Fable chain of thought: "4nln²p/p ≤ n/(4p³): 16p²ln²p ≤ 1: NEVER!!!! THE LADDER IS NEVER THE BEST?!?! Let me recheck the earlier example: p = √(log n) hmm earlier I computed p≈√log n, L≈n/log n: new ≈ √(p³nL′) = (((log n)^{3/4}·n/log n)^{1/2}? hmm: p³ = log^{1.5}n; nL′ = n²/log n: product p³nL′ = n²log^{0.5}n: sqrt: nlog^{0.25}n: TIMES 2: ≈ 2n log^{1/4}n > n!!! WORSE than product!! Earlier I wrote √(p³nL) = n/log^{1/4}n” — ARITHMETIC ERROR (I divided instead of multiplied). Let me recompute: p^{3/2}= log^{3/4}n; √(nL′) = √(n²/log n) = n/log^{1/2}n: product: n·log^{3/4-1/2} = n log^{1/4}n. Yes: WORSE than product. So the ladder construction is DOMINATED. Damn!! OK let me recheck the optimization — maybe I set up"

ALT Fable chain of thought: "4nln²p/p ≤ n/(4p³): 16p²ln²p ≤ 1: NEVER!!!! THE LADDER IS NEVER THE BEST?!?! Let me recheck the earlier example: p = √(log n) hmm earlier I computed p≈√log n, L≈n/log n: new ≈ √(p³nL′) = (((log n)^{3/4}·n/log n)^{1/2}? hmm: p³ = log^{1.5}n; nL′ = n²/log n: product p³nL′ = n²log^{0.5}n: sqrt: nlog^{0.25}n: TIMES 2: ≈ 2n log^{1/4}n > n!!! WORSE than product!! Earlier I wrote √(p³nL) = n/log^{1/4}n” — ARITHMETIC ERROR (I divided instead of multiplied). Let me recompute: p^{3/2}= log^{3/4}n; √(nL′) = √(n²/log n) = n/log^{1/2}n: product: n·log^{3/4-1/2} = n log^{1/4}n. Yes: WORSE than product. So the ladder construction is DOMINATED. Damn!! OK let me recheck the optimization — maybe I set up"

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I enjoyed this conversation with Ryan!
Ryan Peterman
@ryanlpetermanJun 29
Ryan Williams (@rrwilliams) is a professor at MIT and the winner of the Gödel Prize in theoretical computer science. I interviewed him all about his work starting by asking him a popular Leetcode question (3 SUM). In this episode: • Solving Leetcode faster than popular "optimal" solutions • SAT problems and solvers • Hot takes on famous open questions • How to pick good research direction Where to watch: • YouTube - youtu.be/AaK1SL2i_4Y • Spotify - open.spotify.com/episode/0JH… • Apple Podcasts - podcasts.apple.com/us/podcas… • Transcript - developing.dev/p/mit-complex… Thank you to the sponsor of this episode for supporting my work: • WorkOS: makes your app Enterprise Ready with easy to use APIs to add SSO, SCIM, RBAC, and more in just a few lines of code, check them out at workos.com/ Chapters: 00:00 - Intro 00:41 - Asking him a popular Leetcode question 03:54 - Doing better than the popular optimal solution 08:26 - Fine grained complexity 17:00 - A severe strengthening of P vs NP 24:38 - SAT problems and solvers 34:51 - Hot takes on famous open questions 46:57 - Simulating space with time 01:01:02 - Why he solves hard problems 01:02:35 - How to pick good research direction 01:07:14 - Technical book recommendations 01:08:31 - Advice for his younger self 01:11:56 - Outro
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I'm very glad this problem is resolved; it bugged me for way too long. (I had known the poly(log log n) dimension reduction in [Wil18] since 2012; eventually I just published what I had. Of course I was trying to refute SETH using it :)) Congrats to all humans and AIs involved!
Barna Saha@B1ar2n3aJun 26
Given a set of n points in d dimensions, the problems of finding the furthest pair and the closest pair require no explanation even to laymen. However, while finding the closest pair has a much more efficient algorithm,
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Ryan Williams @rrwilliams.bsky.social retweeted
Lijie Chen
@wjmzbmr1Jun 27
This is incredible!!! My first paper in my PhD back in 2018 was on this problem (getting a 2^{logstar n} dim bound); since then, I have been thinking about it on and off for at least 7 years. It's really crazy how things turned out. Last month at OpenAI, our model disproved the famous unit distance conjecture, and the number-theoretic techniques are EXACTLY what you need to settle this furthest pair problem!!! I am so excited about the future of TCS and can't wait to see more open problems I care about being resolved by collaboration between AI and humans :)
Barna Saha@B1ar2n3aJun 26
Given a set of n points in d dimensions, the problems of finding the furthest pair and the closest pair require no explanation even to laymen. However, while finding the closest pair has a much more efficient algorithm,
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Ryan Williams @rrwilliams.bsky.social retweeted
Lijie Chen
@wjmzbmr1May 21
1/ Today, an internal @OpenAI model has refuted Erdős’s unit distance conjecture — a research result that one could recommend “acceptance without any hesitation” to the Annals of Mathematics, one of the most prestigious journals in mathematics. We came across it in a side quest to push our model on the hardest problems.
OpenAI
@OpenAIMay 20
Today, we share a breakthrough on the planar unit distance problem, a famous open question first posed by Paul Erdős in 1946. For nearly 80 years, mathematicians believed the best possible solutions looked roughly like square grids. An OpenAI model has now disproved that belief, discovering an entirely new family of constructions that performs better. This marks the first time AI has autonomously solved a prominent open problem central to a field of mathematics.
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Oh dear. It just won't stop 😢
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I got an email just like this. They were offering me $150k. What is wrong with people?
Rasmus Nielsen@ras_nielsen15 Dec 2025
Bribery to accept graduate students has now been formalized through a company that will help facilitate the transaction.😳
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