There’s a salesman which is employed by large corporation. He have a set of $ n$ cities to visit, connected by roads, forming a graph. But as travel takes a lot of time, he has to pick hotels between visits. And he cannot take any hotel he wishes, there’s is precisely set list on $ m$ hotels.
He have to plan his travel in such way that after visiting $ p$ cities, he have to visit hotel. We may generalise it to say he have to visit $ q$ different hotels ( maybe it will be traveling celebrity problem?).
So basically he has a graph G(E,V), where E-edges, V-nodes, and two sets of nodes: C- cities, H-hotels. $ C \Union H = V$ and $ |C|=n$ , $ |H|=m$ . Find a path in a graph starting at one of the C nodes, ending at different C node and forming pattern$ (p,q)$ , $ p$ nodes from set C, then $ q$ nodes from set H, then repeat. It may not visit every H element, it may visit some of H elements many times, but it have to visit every C once.
So it is as finding Hamilton path but with “rests”.
- Does this problem have a name? Or it is something new?
- In what case it have solution? It probably depends both on numbers p, q, and where on the graph they are located.
- How to find the shortest way discarding hotel costs?
- What is an way to find optimal solution ( cheapest travel) if every hotel cost is the same.
- What is the optimal solution when costs of hotels are different.
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