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Complexity of integer 3SUM

Integer 3SUM asks whether three input integers sum to zero. For polynomially bounded numbers in the specified word-RAM model, the retained target is deterministic time O(n^1.99896), including the added cost analysis and reductions.

Why it matters and what counts as progress

Why it matters: It refines the running-time exponent within an existing subquadratic framework; the incremental exponent improvement is separate from the earlier framework’s hypothesis implications.

What counts as progress: Lower the integer-3SUM exponent in the same word-RAM model, including the cost of arithmetic and reductions for polynomially bounded inputs.

How the problem developed

  1. 1995 · Hardness framework [1]

    Anka Gajentaan and Mark H. Overmars Introduced 3SUM-hardness reductions for geometric problems and recorded the quadratic-time baseline for 3SUM.

  2. 3 Apr 2014 · Algorithmic improvement [2]

    Allan Grønlund and Seth Pettie Gave a deterministic slightly-subquadratic 3SUM algorithm in the real-RAM model, improving the quadratic exponent only by logarithmic factors.

  3. 5 Oct 2026 · Subquadratic algorithm [3]

    Josh Alman and Virginia Vassilevska Williams Reported a deterministic O(n^1.9992)-time algorithm for integer 3SUM on polynomially bounded inputs in the word-RAM model.

  4. 8 Oct 2026 · Exponent refinement [4]

    Claude-assisted follow-up project Reported a further deterministic O(n^1.99896) bound under the same input and word-RAM assumptions.

AI contributions 2

Truly Subquadratic 3SUM and Truly Subcubic APSP via Triangles in Sparse Lopsided Graphs

Josh Alman, Virginia Vassilevska Williams

Claude’s original sparse-output thin-matrix multiplication algorithm, as presented in Section 2 and related to the certified 3SUM/APSP theorems. Human-optimized numerical parameters, new reductions, data structures and Hinted OMv extensions retain human credit.

AI involvementAI-autonomous
ImportanceMajor
Technical verificationFormal (Lean)
Human understandingFull
Paper date:

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Sources

  1. Gajentaan and Overmars · On a class of O(n²) problems in computational geometry (1995)
  2. Grønlund and Pettie · Threesomes, degenerates, and love triangles (2014)
  3. Alman–Vassilevska Williams: algorithms and methodology
  4. Improved-exponents project and contribution scope
  5. Completed Lean solution module
  6. Comparator check log
  7. Formalization project
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Publication dates and editorial changes.