Chinese Postman Problem: finding best connections between odd-degree nodes
As remarked in the comments, Wikipedia gives a reduction from route inspection to minimum-weight matchings.  Vladimir Kolmogorov has published a fast implementation of the weighted version of Edmonds' blossom algorithm, in C++&nbsp;[1].

[1] V. Kolmogorov, Blossom V: A new implementation of a minimum cost perfect matching algorithm. Mathematical Programming Computation, 1(1):43&ndash;67, 2009.