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

Research dossier · v1 · updated 2026-10-30 · 3,308 words · 26 sources · 27 claims · verification: verified

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.

## Contents

- Executive summary
- Background and core concepts
- Timeline of key milestones
- State of the art
  - Shortest paths beyond the sorting barrier
  - Maximum flow and minimum-cost flow
  - Matrix multiplication exponent
  - Time-space simulation and catalytic computing
  - Other 2026 preprints
  - Quantitative benchmarks
- Open problems and active debates
- Major programs and players
- Outlook: what to watch (coming years)
- Key papers
- 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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