Find the shortest path between a given source and a set of destinations

问题

You are given a weighted connected graph (20 nodes) with all edges having positive weight. We have a robot that starts at point A and it must pass at points B, D and E for example. The idea is to find the shortest path that connects all these 4 points. The robot also has a limited battery, but it can be recharged in some points.

After researching on the internet I have two algorithms in mind: Dijkstra's and TSP. Dijkstra's will find the shortest path between a node and every other node and TSP will find the shortest path that connects all points. Is there any variant of the TSP that only finds the shortest path between a set of nodes? After all, in the TSP all nodes are labeled "must-pass". I'm still not taking in account the battery constraint.

Thanks in advance!

回答1:

You can reduce your graph to a TSP and then invoke a TSP algorithm on it:

1. Use Floyd-Warshall algorithm to find the distance u,v for ALL pairs of vertices u and v.
2. Create a new graph, containing only the "desired" vertices, and set the weight between two such vertices u and v as the distance found by Floyd-Warshall.
3. Run TSP Solver on the modified graph to get the path in the modified graph, and switch each edge in the modified graph with a shortest path from the original graph.

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The above is optimal, because assume there is a shorter path.

D0=u->...D1->...->D2->...->Dk->...->t=D{k+1}

Di->...->D{i+1} has at least the weight of FloydWarshall(Di,D{i+1}) (correctness of Floyd-Warshall), and thus the edges (D0,D1),(D1,D2),...,(Dk,D{k+1) exist in the modified graph with a weight smaller/equal the weight in the given path.

Thus, from correctness of your TSP-Solver, by using D0->D1->...->Dk->D{k+1}, you get a path that is at least as good as the candidate optimal path.

回答2:

You might also want to look into the generalized traveling salesman problem (GTSP): The nodes are partitioned into subsets, and the problem is to find the minimum-length route that visits exactly one node in each subset. The model is allowed to choose whichever node it wants from each subset. If there are nodes that must be visited, you can put them in a subset all by themselves.

来源：https://stackoverflow.com/questions/28925145/find-the-shortest-path-between-a-given-source-and-a-set-of-destinations