Alman and Williams Break the n² and n³ Barriers for 3SUM and APSP
Josh Alman and Virginia Vassilevska Williams have posted a preprint, "Truly Subquadratic 3SUM and Truly Subcubic APSP via Triangles in Sparse Lopsided Graphs," delivering the first-ever polynomial-level improvements on two textbook problems: for n polynomially-bounded integers, 3SUM can be solved deterministically in O(n^1.9992) time, and APSP with polynomial weights in O(n^2.9995) time. These break through the long-standing n² and n³ barriers respectively, effectively refuting the 3SUM conjecture and the APSP conjecture.
Confirmed
- The paper presents deterministic algorithms, the first truly subquadratic/subcubic results for 3SUM and APSP
- The 3SUM algorithm comes with a Lean formalized proof (according to theoretical computer scientist Ilya Razenshteyn)
- Posts circulating suggest Anthropic's internal model Claude was involved in discovering the subquadratic algorithm; @AMBNNJ called it an independent discovery by an internal model
Why it matters
- The 3SUM and APSP conjectures are pillars of fine-grained complexity theory, underpinning a large body of conditional lower bounds; their refutation redraws the landscape of the field
- @thegautamkamath relayed comments calling it "a genuinely extraordinary result" that nonetheless "takes away much of the fun of the field" — many classic hardness assumptions are now shaken
- If Claude's involvement is confirmed, it would be a landmark case of AI achieving a substantive breakthrough in open-ended theoretical research
2026-10-06 ~ 2026-10-06 · 6 related posts
Primary sources
- Anthropic's internal model found the first truly subquadratic 3SUM algorithm, preprint claims — AMBNNJ ·
- 3SUM Solved in O(n^1.9992): Alman & Vassilevska Williams Refute the 3SUM and APSP Hypotheses — auto_grad_ ·
- Claude produces O(n^1.9992) 3SUM algorithm with Lean proof, vetted by top experts — thegautamkamath ·
- Truly subcubic APSP and truly subquadratic 3SUM breakthrough reshapes fine-grained complexity — thegautamkamath · 2026-10-06
- [source] Claude produces O(n^1.9992) 3SUM algorithm with Lean proof, vetted by top experts — thegautamkamath · 2026-10-06
- [source] 3SUM Solved in O(n^1.9992): Alman & Vassilevska Williams Refute the 3SUM and APSP Hypotheses — auto_grad_ · 2026-10-06
- [source] Anthropic's internal model found the first truly subquadratic 3SUM algorithm, preprint claims — AMBNNJ · 2026-10-06
- New arXiv paper refutes 3SUM and APSP hypotheses; Claude credited with subquadratic algorithm — burny_tech · 2026-10-06
1 near-duplicate retellings: ctjlewis