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 $2^n n^2 / 2^{\Omega(\sqrt{\log n})}$. We achieve this by strategically remodeling the dynamic programming recursion as a min-plus matrix product, for which faster-than-na\"ive algorithms exist.

翻译：由Bellman以及独立工作的Held与Karp提出的旅行商问题动态规划解法运行时间为$O(2^n n^2)$，且在最近六十年内未有改进。我们首次突破了这一屏障，设计出运行在确定性时间$2^n n^2 / 2^{\Omega(\sqrt{\log n})}$内的算法。通过将动态规划递归策略性地重构为最小-加矩阵乘积（对此存在比朴素算法更快的算法），我们实现了这一突破。