{"slug":"algorithms-complexity","topic":"algorithms-complexity","title":"Algorithms & Complexity Breakthroughs: State of the Field (October 2026)","summary":"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.","version":1,"published_at":"2026-10-30","updated_at":"2026-10-30","stats":{"words":3308,"citations":26,"claims":27,"timeline":15,"benchmarks":14,"open_problems":5,"key_papers":14,"glossary":13},"verification":{"status":"verified","method":"automated figure and citation-relevance triage against cited abstracts, plus targeted AI review with web checks of flagged statements; automated citation resolution","claims_checked":12,"issues_fixed":7,"verified_at":"2026-10-30"},"ai_assisted":true,"price":{"usd":"0.20","protocol":"x402","network":"eip155:84532","asset":"USDC"},"url":"https://space-kitty.com/dossiers/algorithms-complexity","preview_url":"https://space-kitty.com/api/v1/dossiers/algorithms-complexity/preview","purchase_url":"https://space-kitty.com/api/v1/dossiers/algorithms-complexity","toc":[{"level":2,"text":"Executive summary","anchor":"executive-summary"},{"level":2,"text":"Background and core concepts","anchor":"background-and-core-concepts"},{"level":2,"text":"Timeline of key milestones","anchor":"timeline-of-key-milestones"},{"level":2,"text":"State of the art","anchor":"state-of-the-art"},{"level":3,"text":"Shortest paths beyond the sorting barrier","anchor":"shortest-paths-beyond-the-sorting-barrier"},{"level":3,"text":"Maximum flow and minimum-cost flow","anchor":"maximum-flow-and-minimum-cost-flow"},{"level":3,"text":"Matrix multiplication exponent","anchor":"matrix-multiplication-exponent"},{"level":3,"text":"Time-space simulation and catalytic computing","anchor":"time-space-simulation-and-catalytic-computing"},{"level":3,"text":"Other 2026 preprints","anchor":"other-2026-preprints"},{"level":3,"text":"Quantitative benchmarks","anchor":"quantitative-benchmarks"},{"level":2,"text":"Open problems and active debates","anchor":"open-problems-and-active-debates"},{"level":2,"text":"Major programs and players","anchor":"major-programs-and-players"},{"level":2,"text":"Outlook: what to watch (coming years)","anchor":"outlook-what-to-watch-coming-years"},{"level":2,"text":"Key papers","anchor":"key-papers"},{"level":2,"text":"Glossary","anchor":"glossary"}],"sample_claims":["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)."],"includes":["markdown","claims","timeline","benchmarks","open_problems","key_papers","glossary","bibliography"],"citation_count":26,"how_to_buy":"GET https://space-kitty.com/api/v1/dossiers/algorithms-complexity with an x402 client ($0.20, USDC on Base Sepolia (testnet))."}