My problem is similar to the problem here https://cs.stackexchange.com/questions/2244/need-a-np-complete-proof-on-an-example , but it is a little different.
Here is my problem:
There are three islands, A, B and C, and a lot of fan-shaped rafts. We must build a bridge from A-->B-->C, and the number of rafts required for each part is already known, say, four rafts are required for connecting AB and three rafts are required for connecting BC.
These rafts are at different positions originally and they can rotate without cost. An interesting thing is that they can overlap with each other if necessary. The distance of moving one raft can be calculated as the distance between the center of mass's original position and its deployed position.
The objective is to find the solution having the minimum total distance of moving rafts in order to have bridge A-->B-->C and using the exact number of rafts for each part of bridge.
I used to the following figure to show my question.
We can see from this figure that the arrangement might not be a straight line, and the rafts can overlap with another one.
There are too many candidates locations for these rafts. It seems the problem is NPC. I do not know whether I am right and how to prove it to be NPC. Anyone know how to solve it? Thanks!