Problem:
Score : 100 points

Problem Statement

There are N rabbits, numbered 1, 2, \ldots, N.

For each i, j (1 \leq i, j \leq N), the compatibility of Rabbit i and j is described by an integer a_{i, j}.
Here, a_{i, i} = 0 for each i (1 \leq i \leq N), and a_{i, j} = a_{j, i} for each i and j (1 \leq i, j \leq N).

Taro is dividing the N rabbits into some number of groups.
Here, each rabbit must belong to exactly one group.
After grouping, for each i and j (1 \leq i < j \leq N), Taro earns a_{i, j} points if Rabbit i and j belong to the same group.

Find Taro's maximum possible total score.

Constraints

All values in input are integers.

1 \leq N \leq 16

|a_{i, j}| \leq 10^9

a_{i, i} = 0

a_{i, j} = a_{j, i}

Input

Input is given from Standard Input in the following format:

N
a_{1, 1} \ldots a_{1, N}
:
a_{N, 1} \ldots a_{N, N}

Output

Print Taro's maximum possible total score.

Sample Input 1

3
0 10 20
10 0 -100
20 -100 0

Sample Output 1

20

The rabbits should be divided as \{1, 3\}, \{2\}.

Sample Input 2

2
0 -10
-10 0

Sample Output 2

0

The rabbits should be divided as \{1\}, \{2\}.

Sample Input 3

4
0 1000000000 1000000000 1000000000
1000000000 0 1000000000 1000000000
1000000000 1000000000 0 -1
1000000000 1000000000 -1 0

Sample Output 3

4999999999

The rabbits should be divided as \{1, 2, 3, 4\}.
Note that the answer may not fit into a 32-bit integer type.

Sample Input 4

16
0 5 -4 -5 -8 -4 7 2 -4 0 7 0 2 -3 7 7
5 0 8 -9 3 5 2 -7 2 -7 0 -1 -4 1 -1 9
-4 8 0 -9 8 9 3 1 4 9 6 6 -6 1 8 9
-5 -9 -9 0 -7 6 4 -1 9 -3 -5 0 1 2 -4 1
-8 3 8 -7 0 -5 -9 9 1 -9 -6 -3 -8 3 4 3
-4 5 9 6 -5 0 -6 1 -2 2 0 -5 -2 3 1 2
7 2 3 4 -9 -6 0 -2 -2 -9 -3 9 -2 9 2 -5
2 -7 1 -1 9 1 -2 0 -6 0 -6 6 4 -1 -7 8
-4 2 4 9 1 -2 -2 -6 0 8 -6 -2 -4 8 7 7
0 -7 9 -3 -9 2 -9 0 8 0 0 1 -3 3 -6 -6
7 0 6 -5 -6 0 -3 -6 -6 0 0 5 7 -1 -5 3
0 -1 6 0 -3 -5 9 6 -2 1 5 0 -2 7 -8 0
2 -4 -6 1 -8 -2 -2 4 -4 -3 7 -2 0 -9 7 1
-3 1 1 2 3 3 9 -1 8 3 -1 7 -9 0 -6 -8
7 -1 8 -4 4 1 2 -7 7 -6 -5 -8 7 -6 0 -9
7 9 9 1 3 2 -5 8 7 -6 3 0 1 -8 -9 0

Sample Output 4

132

Solution in C++ (GCC 9.2.1) (judge verdict: Time Limit Exceeded)

#include <bits/stdc++.h>
using namespace std;

const int mxN = 16, mxV = (1 << 16) - 1;
int n, comp[mxN][mxN];
long long dp[mxV];
set<int> used;

long long score(vector<int>& a)
{
    long long val = 0;
    int l = a.size();
    for (int i = 0; i < l; i++)
    {
        for (int j = i + 1; j < l; j++)
        {
            val += comp[a[i]][a[j]];
        }
    }
    return val;
}

long long recurse(int mask)
{
    if (mask == 0)
        return 0;
    if (used.count(mask) > 0)
        return dp[mask];
    
    // cout << mask << endl;

    vector<int> ones;
    for (int i = 0; i < n; i++)
    {
        if ((mask & (1 << i)) > 0)
        {
            ones.push_back(i);
        }
    }
    // for (auto& n : ones)
    // {
    //     cout << n << " ";
    // }
    // cout << endl;
    int smallest = ones[0];
    // cout << "smallest: " << smallest << endl;
    mask ^= (1 << smallest);
    // cout << "mask now: " << mask << endl;

    long long ans = -1e15;
    int b = 0;
    do
    {
        vector<int> cur; cur.push_back(smallest);
        for (int i = 0; i < n; i++)
        {
            if ((b & (1 << i)) > 0)
                cur.push_back(i);
        }
        int tmp = mask ^ b;

        ans = max(ans, recurse(tmp) + score(cur));

        // process subset b
    } while (b = (b - mask) & mask);

    dp[mask] = ans;
    used.insert(mask);
    return ans;
}

int main()
{
    cin >> n;
    for (int i = 0; i < n; i++)
    {
        for (int j = 0; j < n; j++)
        {
            cin >> comp[i][j];
        }
    }

    cout << recurse((1 << n) - 1);
}