Chinese postman problem, but where the postman have to visit each vetex at least once.
Is there a name for this problem?
What is the ideal algorithm to solve this problem?
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Let $G$ be an edge-weighted graph. Create a new graph $G'$ in which the weight of the edge $(x,y)$ is the cost of the shortest path between $x$ and $y$ in $G$. The problem you are interested in is the traveling salesman problem (TSP) on $G'$.