The dynamic programming solution to the traveling salesman problem due to Bellman, and independently Held and Karp, runs in time $O(2^n n^2)$, with no improvement in the last sixty years. We break this barrier for the first time by designing an algorithm that runs in deterministic time $2^n n^2 / 2^{Ω(\sqrt{\log n})}$. We achieve this by strategically remodeling the dynamic programming recursion as a min-plus matrix product, for which faster-than-naïve algorithms exist.
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