Valued permutation

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Problem: there are 2 parallel arrays of positive values A and B of size n.

How to find the minimal value for the following target function:

F(A, B) = Ak + Bk * F(A', B')

where A', B' denote the arrays A and B with their k:th element removed.

I was thinking about dynamic programming approach, but with no success.

How to apply on such kind of problems, where we need to evaluate given function on a permutation?

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