Route inspection problem
In graph theory, a branch of mathematics and computer science, the Chinese postman problem (CPP), postman tour or route inspection problem is to find a shortest closed path or circuit that visits every edge of a (connected) undirected graph. When the graph has an Eulerian circuit (a closed walk that covers every edge once), that circuit is an optimal solution. Otherwise, the optimization problem is to find the smallest number of graph edges to duplicate (or the subset of edges with the minimum possible total weight) so that the resulting multigraph does have an Eulerian circuit. It may be solved in polynomial time.
The problem was originally studied by the Chinese mathematician Kwan Mei-Ko in 1960, whose Chinese paper was translated into English in 1962. The alternative name "Chinese postman problem" was coined in his honor; different sources credit the coinage either to Alan J. Goldman or Jack Edmonds, both of whom were at the U.S. National Bureau of Standards at the time.
The undirected route inspection problem may be solved in polynomial time by an algorithm based on the concept of a T-join. Let T be a subset of the vertex set of a graph. An edge set J is called a T-join if the collection of vertices that have an odd number of neighboring edges in J is exactly the set T. A T-join exists whenever every connected component of the graph contains an even number of vertices in T. The T-join problem is to find a T-join with the minimum possible number of edges or the minimum possible total weight.
For any T, a smallest T-join (when it exists) necessarily consists of |T| paths that join the vertices of T in pairs. The paths will be such that the total length or total weight of all of them is as small as possible. In an optimal solution, no two of these paths will share any edge, but they may have shared vertices. A minimum T-join can be obtained by constructing a complete graph on the vertices of T, with edges that represent shortest paths in the given input graph, and then finding a minimum weight perfect matching in this complete graph. The edges of this matching represent paths in the original graph, whose union forms the desired T-join. Both constructing the complete graph, and then finding a matching in it, can be done in O(n3) computational steps.
...
Wikipedia