# Anthropic model refutes 3SUM and APSP hypotheses with subquadratic algorithms

Digest AI · Research · published 2026-10-06T07:16:54Z

Canonical: https://digestai.news/story/anthropic-model-refutes-3sum-and-apsp-hypotheses-with-subquadratic-alg

## Summary

Anthropic’s internal research model has produced the first deterministic algorithms that break the long‑standing quadratic and cubic barriers for two classic problems. The new method solves 3SUM on n integers in O(n^{1.9992}) time and all‑pairs shortest paths on directed n‑vertex graphs with polynomially bounded integer weights in O(n^{2.9995}) time. By exploiting a thin‑matrix product technique—computing selected entries of an N×N product in O(N^2/D^{0.063}) operations with D ≤ N^{1/18}—the work refutes the 3SUM and APSP hypotheses and, via known reductions, also overturns related conjectures such as Exact Triangle, Zero‑Weight k‑Clique, and several online matrix‑vector hypotheses.

The authors describe the algorithm as a modification of Coppersmith’s rectangular matrix multiplication, using a ten‑multiplication identity of Schönhage to limit work to only the needed entries. They also present a data‑structure version that can answer single‑entry queries on the fly. The results are posted as a pre‑print on arXiv, with accompanying code and data for community verification.

## Key points

- Anthropic’s internal model solves 3SUM in O(n^{1.9992}) time, beating the quadratic barrier
- The same technique yields APSP on directed graphs in O(n^{2.9995}) time, refuting the APSP hypothesis
- The approach uses a thin‑matrix product algorithm with D ≤ N^{1/18}, achieving O(N^2/D^{0.063}) operations for selected entries

## Why it matters

Proving subquadratic 3SUM and subcubic APSP challenges core complexity assumptions, opening new avenues for faster algorithms across computer science and AI research.

## Sources

1. [Internal Anthropic model refuted the 3SUM and APSP hypotheses: first truly subquadratic 3SUM and truly subcubic APSP (Alman & Vassilevska Williams)](https://arxiv.org/abs/2610.06783) (arxiv.org, 2026-10-06, primary source)

## Cite

Digest AI, "Anthropic model refutes 3SUM and APSP hypotheses with subquadratic algorithms", 6 October 2026, https://digestai.news/story/anthropic-model-refutes-3sum-and-apsp-hypotheses-with-subquadratic-alg

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