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Research dossier · version 1 · updated Oct 30, 2026

Algorithms & Complexity Breakthroughs: State of the Field (October 2026)

Covers near-linear-time maximum flow and shortest-path algorithms, the 2025 breach of the sorting barrier for directed shortest paths, the matrix multiplication exponent, and time-space simulation results as of October 2026. The key takeaway: the best matrix multiplication exponent bound is a 2026 preprint, and a much stronger claimed bound is still unrefereed.

3,308 words26 sources27 evidence-mapped claims14 benchmarks15 timeline events✓ citation-verified Oct 30, 2026

AI-assisted synthesis of the cited literature, independently fact-checked against its sources; not peer reviewed.

Contents

  1. Executive summary
  2. Background and core concepts
  3. Timeline of key milestones
  4. State of the art
  5. Shortest paths beyond the sorting barrier
  6. Maximum flow and minimum-cost flow
  7. Matrix multiplication exponent
  8. Time-space simulation and catalytic computing
  9. Other 2026 preprints
  10. Quantitative benchmarks
  11. Open problems and active debates
  12. Major programs and players
  13. Outlook: what to watch (coming years)
  14. Key papers
  15. Glossary

Sample findings

  • Exact maximum flow and minimum-cost flow on directed graphs with polynomially bounded integer data can be computed in m^(1+o(1)) time.
  • The almost-linear max flow and min-cost flow result has a peer-reviewed journal version in JACM (2025).
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