In order to minimize the sum of ~V_{1 \dots N}~, we'll want to generally use the smallest ~V~ values possible. After some thought, it should become clear that very small values will indeed do the trick – in fact, limiting ourselves to just ~1 \le V_i \le 3~ is always sufficient to produce a valid solution. However, upon further thought, we realize that those values may not quite yield an optimal solution – in particular, it may be more optimal overall to use various products of small primes. For example, an instance of ~V_i = 6~ could allow many other ~V~ values to be reduced from ~3~ to ~2~. It's difficult to say exactly how large the ~V~ values in an optimal solution might get for a given value of ~N~, but limiting ourselves to all possible values between ~1~ and ~30~ (inclusive) will be more than enough.

That insight isn't enough to directly suggest how the ~V~ values should optimally be assigned, but at that point we can use dynamic programming to efficiently find an optimal assignment. Let's represent the system of terminals and wires as a tree, rooted at an arbitrary node such as node ~1~. Then, let ~DP[i][v]~ be the minimum sum of ~V~ values in the subtree rooted at ~i~, such that ~V_i = v~. ~DP[i][v]~ can be computed recursively based on the ~DP~ values of ~i~'s children. Each child ~c~ can be handled independently, and we can simply consider all possible choices for ~V_c~ such that the GCD of ~V_i~ and ~V_c~ satisfies the wire connecting terminals ~i~ and ~c~, choosing the one which minimizes ~DP[c][V_c]~. At the end, the answer will be ~\min\{DP[1][v]\}~ over all possible values of ~v~. The time complexity of this algorithm is ~\mathcal O(N)~.