3SUM and APSP hypotheses refuted as subquadratic breakthrough lands
aran_nayebi · x · 2026-10-06
A rare day in theoretical CS: Josh Alman and Virginia Vassilevska Williams posted an arXiv paper giving the first polynomial improvements over textbook algorithms — deterministic 3SUM in O(n^1.9992) and APSP in O(n^2.9995) — refuting the 3SUM and APSP hypotheses and, via known reductions, several related conjectures. The key is a new thin matrix product algorithm built by modifying Coppersmith's rectangular matrix multiplication. The KLS conjecture was also announced as proven the same day.
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