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++14 (GCC 5.4.1) (judge verdict: Accepted)

#include<iostream>
#include<cstring>
#include<algorithm>
#include<cstdio>
#include<vector>
#include<set>
#include<map>
#include<string>
 
using namespace std;
 
	long long dp[1<<16];
	int nr[1<<16];
	long long a[17][17];
	long long sum[1<<16];
	
	//long long pot16[16];
	
	string binarnizapis(int a){
		string s = "";
		for(int i = 0; i < 16; i++){
			if(a&(1<<i)) s+="1";
			else s+="0";
		}
		
		return s;
		
	}
	vector < int > veki;
 
int main(){
	
	for(int i = 0; i < (1<<16); i++)
		for(int j = 0; j < 16; j++)
			if(i&(1<<j)) nr[i]++;
 
	int n; cin >> n;
	for(int i = 0; i < n; i++)
		for(int j = 0; j < n; j++){
			cin >> a[i][j];
		}
		
	for(int i = 0; i < (1<<16); i++){
		for(int j = 0; j < 16; j++){
			for(int k = 0; k < j; k++){
				if(i&(1<<j) && i&(1<<k)) sum[i] += a[j][k];
			}
		}
		dp[i] = sum[i]; // sve za jednu grupu
	}	
	
	
//	cout << "pass" << endl;
		
		
	for(int j = 1; j < (1<<n); j++){
		
		
			
	//	cout << "-----------------------------" << endl;	
			
		veki.clear();
		for(int k = 0; k < n; k++)	if((j&(1<<k)) == 0) veki.push_back(k);
		
			
		for(int k = 1; k < (1<<(n - nr[j])); k++){
			
			int c_j = j;
			int idx = 0;
				
			for(int i = 0; i < veki.size(); i++){
				
				if(k&(1<<i)) c_j +=(1<<veki[i]);					
						
			}
			
	//		cout << k << endl;
	//		cout << binarnizapis(c_j) << " " << binarnizapis(j) << " " << binarnizapis(c_j^j) << endl;
	//		cout << dp[j] + dp[c_j^j] << " " << dp[c_j] << endl;
	//		system("pause");	
			
			dp[c_j] = max(dp[c_j],dp[j] + sum[c_j^j]);
					
		}
			
	}
	
	cout << dp[(1<<n)-1];
		
		
		
			
 
//	long long sum = 0;
//	for(int i = 0; i < (1<<16); i++){
//		sum += (1<<(16-nr[i]));
//	}
//
//	pot16[1] = 16;
//	for(int i = 2; i < 16; i++){
//		pot16[i] = 16*pot16[i-1]/i;
//	}
//
//	for(int i = 16; i >= 0; i--){
//		sum += (1<<i)*pot16[16-i];
//	}
//	
//	cout << sum << endl; 
 
 
 
 
 
	return 0;
}