The Graphhopper Java Sample Code by Graphhopper presents how API communication can access routing and logistics features to get from point A to B. Solution API test and Vrp API test are available on GitHub.
The Google Maps API naturally has a rich history on ProgrammableWeb. Over the last year or so, Google has released additional maps-related APIs as separate services. One of the latest ones is the Google Distance Matrix API. This allows a developer to submit a matrix of locations, as origins and destinations, and get a list of the distances in return.