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So we can solve the traveling salesman problem using dynamic programming.  Subproblems will be subsets of the vertices $S$ from ${1,2,....,n}$ $(1,2,....,n)$  that includes the origin and a specific node j. Let $C(S,j)$ be the length of the shortest path visiting each node in S starting at 1 and stopping at $j$. So $C(S,j)$ = $min(C(S-j,i)+d_{ij})$ for each $i$, where $i$ are all the neighbors of $j$.