(Analysis by Nathan Pinsker)

Although $k$ is rather small, trying all possible positions for the doors is $O(\binom n k)$ and is still way too slow.

Sometimes it's helpful in solving a problem to think about simpler cases of the problem first. When you're given an optimization problem on a circular array, one easy simplification is to think about it in a linear setting instead. So, if Farmer John's barn was linear, then how could we solve it? For starters, we clearly need to put a barn at the left-most position. It's more apparent now that we can do dynamic programming, since if we iterate from left to right, the only state that we need to keep track of is the number of exterior doors used so far, and the position of the last exterior door. This algorithm would have a running time of $O(n^2k)$, which is quite fast -- looking at the bounds, we can actually introduce an extra factor of $n$ when transferring this to the circular case without running over the time limit.

Since we have a lot of running time to spare, we can simply iterate over all possible locations of the first interior door. This turns the problem into effectively a linear one, although a bit of care is needed when processing the last door. We obtain an $O(n^3k)$ algorithm, which is fast enough to receive full credit.

Here's Mark Gordon's solution:

#include <iostream>
#include <vector>
#include <algorithm>
#include <cstring>

using namespace std;

#define MAXN 1010
#define MAXK 10
#define INF 0x3FFFFFFFFFFFFFFFLL

int N, K;
long long A[MAXN];
long long DP[MAXK][MAXN];

int main() {
cin >> N >> K;
for (int i = 0; i < N; i++) {
cin >> A[i];
}

long long result = INF;
for (int i = 0; i < N; i++) {
memset(DP, 0x3F, sizeof(DP));
DP[0][N] = 0;

for (int k = 1; k <= K; k++) {
for (int j = 0; j < N; j++) {
long long val = 0;
for (int a = j + 1; a <= N; a++) {
val += A[a - 1] * (a - j - 1);
DP[k][j] = min(DP[k][j], val + DP[k - 1][a]);
}
}
}
result = min(result, DP[K][0]);
rotate(A, A + 1, A + N);
}
cout << result << endl;
}