A team of ten Claude Opus 5.5 agents has developed a new algorithm, named C-HD, that achieves a formally verified improvement over published asymptotic bounds for the shortest path problem. The research, conducted over 15 hours using a message-board-based collaborative setup, focused on finding exact shortest-path distances in directed graphs with non-negative real weights.
AI Agents Discover Faster Shortest Path Algorithm and Provide Formal Lean Proof
The C-HD algorithm provides a better asymptotic upper bound in specific regimes. For instance, on graphs where the number of edges m approx n log^{3/4} n, the algorithm achieves a runtime of O(n log^{11/12} n), improving upon the O(n log n) bound of Dijkstra’s algorithm. While this represents a mathematical improvement in the asymptotic limit, the researchers noted that the constants involved in the formal construction are enormous, and the results do not yet establish a practical speedup in real-world applications.
The correctness and efficiency of C-HD were formally verified using the Lean theorem prover. The verification process included a complete project build and multiple kernel replays to ensure the theorem dependencies and the algorithm's properties were strictly maintained. To ensure robustness, the agents also implemented a Bellman–Ford fallback to handle inputs outside the certified density range.
Sources
- A Faster Shortest Path Algorithm （Hacker News Frontpage, 2026-09-22）
- Lean proof