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# (.NET Core 3.1.201) (judge verdict: Accepted)

using System;
using System.Collections.Generic;
using System.Linq;
using System.IO;
using System.Text;
using System.Numerics;
using System.Threading;
using System.Runtime.CompilerServices;
using System.Diagnostics;
using static System.Math;
using static System.Array;
using static AtCoder.Cout;
using static AtCoder.Tool;
using static AtCoder.ModInt;
namespace AtCoder
{
    class AC
    {
        const int MOD = 1000000007;
        //const int MOD = 998244353;
        const int INF = int.MaxValue / 2;
        const long SINF = long.MaxValue / 3;
        const double EPS = 1e-8;
        static readonly int[] dI = { 0, 1, 0, -1, 1, -1, -1, 1 };
        static readonly int[] dJ = { 1, 0, -1, 0, 1, 1, -1, -1 };
        static void Main(string[] args)
        {
            //var sw = new StreamWriter(Console.OpenStandardOutput()) { AutoFlush = false }; Console.SetOut(sw);

            var th = new Thread(Run, 1 << 26);
            th.Start();
            th.Join();

            //Run();
            Console.Out.Flush();
        }
        static void Run()
        {
            int Testcase = 1;
            //Testcase = Cin.Int;
            for (var _ = 0; _ < Testcase; _++) Solve();
        }
        static void Solve()
        {
            int N = Cin.Int;
            var a = new long[N][];
            for (var i = 0; i < N; i++) a[i] = Cin.ReadSplitLong;
            var dp = new long[(1 << N)];
            for(var i = 0; i < (1 << N); i++)
            {
                long def = 0;
                for (var j = 0; j < N; j++) for (var k = j + 1; k < N; k++)
                    {
                        if (Bit(i, j) && Bit(i, k)) def += a[j][k];
                    }
                dp[i] = def;
                for(var j = i; j > 0; j = (j - 1) & i)
                {
                    dp[i] = Max(dp[i], dp[j] + dp[i ^ j]);
                }
            }
            OutL(dp[(1 << N) - 1]);
        }

        public struct Edge
        {
            public int from;

            public int to;
            public long dist;

            public Edge(int t, long c)
            {
                from = -1;
                to = t;
                dist = c;
            }
            public Edge(int f, int t, long c)
            {
                from = f;
                to = t;
                dist = c;
            }
        }
    }
    struct ModInt
    {
        public long value;
        private const int MOD = 1000000007;
        //private const int MOD = 998244353;
        public ModInt(long value) { this.value = value; }
        public static implicit operator ModInt(long a)
        {
            var ret = a % MOD;
            return new ModInt(ret < 0 ? (ret + MOD) : ret);
        }
        public static ModInt operator +(ModInt a, ModInt b) => (a.value + b.value);
        public static ModInt operator -(ModInt a, ModInt b) => (a.value - b.value);
        public static ModInt operator *(ModInt a, ModInt b) => (a.value * b.value);
        public static ModInt operator /(ModInt a, ModInt b) => a * Modpow(b, MOD - 2);

        public static ModInt operator <<(ModInt a, int n) => (a.value << n);
        public static ModInt operator >>(ModInt a, int n) => (a.value >> n);
        public static ModInt operator ++(ModInt a) => a.value + 1;
        public static ModInt operator --(ModInt a) => a.value - 1;
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public static ModInt Modpow(ModInt a, long n)
        {
            if (n < 0) return Modpow(Modpow(a, -n), MOD - 2);
            var k = a;
            ModInt ret = 1;
            while (n > 0)
            {
                if ((n & 1) != 0) ret *= k;
                k *= k;
                n >>= 1;
            }
            return ret;
        }
        private static readonly List<long> Factorials = new List<long>() { 1 };
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public static ModInt Fac(long n)
        {
            for (var i = Factorials.Count(); i <= n; i++)
            {
                Factorials.Add((Factorials[i - 1] * i) % MOD);
            }
            return Factorials[(int)n];
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public static ModInt nCr(long n, long r)
        {
            if (n < 0 || r < 0) return 0;
            return n < r ? 0 : Fac(n) / (Fac(r) * Fac(n - r));
        }
        public static explicit operator int(ModInt a) => (int)a.value;
    }
    public class Rolling_Hash
    {
        const ulong m30 = (1UL << 30) - 1;
        const ulong m31 = (1UL << 31) - 1;
        const ulong MOD = (1UL << 61) - 1;
        const ulong Pl = (MOD << 1) << 1;
        private uint B;
        private string S;
        ulong[] hash;
        ulong[] pw;

        public Rolling_Hash(string str)
        {
            S = str;
            B = (uint)new Random().Next(1 << 12 + 1, int.MaxValue);
            int L = S.Length;
            hash = new ulong[L + 1];
            pw = new ulong[L + 1];
            hash[0] = 0;
            pw[0] = 1;
            for (var i = 0; i < L; i++)
            {
                hash[i + 1] = CalcMod(Mul(hash[i], B) + S[i]);
                pw[i + 1] = CalcMod(Mul(pw[i], B));
            }
        }

        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public ulong GetHashValue(int idx) => hash[idx];
        [MethodImpl(MethodImplOptions.AggressiveInlining)]//segment [l,r]
        public ulong Hash_fold(int l, int r) => CalcMod(Pl + hash[r + 1] - Mul(hash[l], pw[r - l + 1]));
        [MethodImpl(MethodImplOptions.AggressiveInlining)]//segment[start,start+len-1]
        public ulong Hash_sub(int start, int len) => CalcMod(Pl + hash[start + len] - Mul(hash[start], pw[len]));
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public ulong[] GetHashArray() => hash;

        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        ulong Mul(ulong a, ulong b)
        {
            ulong au = a >> 31;
            ulong ad = a & m31;
            ulong bu = b >> 31;
            ulong bd = b & m31;
            ulong mid = ad * bu + au * bd;
            ulong midu = mid >> 30;
            ulong midd = mid & m30;
            return au * bu * 2 + midu + (midd << 31) + ad * bd;
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        ulong CalcMod(ulong x)
        {
            ulong xu = x >> 61;
            ulong xd = x & MOD;
            ulong res = xu + xd;
            if (res >= MOD) res -= MOD;
            return res;
        }
    }
    public class Priority_Queue<T>
    {
        private List<T> Q;
        private readonly Comparison<T> Func_Compare;
        public Priority_Queue(Comparison<T> comp)
        {
            Func_Compare = comp;
            Q = new List<T>();
        }
        private void PushHeap(List<T> list, T item)
        {
            int n = list.Count();
            list.Add(item);

            while (n != 0)
            {
                int pIndex = (n - 1) / 2;

                if (Func_Compare(list[n], list[pIndex]) < 0)
                {
                    Swap(Q, n, pIndex);
                }
                else { break; }

                n = pIndex;
            }
        }
        private void PopHeap(List<T> list)
        {
            int n = list.Count() - 1;
            list[0] = list[n];
            list.RemoveAt(n);

            int cur = 0;
            int comp;

            while (2 * cur + 1 <= n - 1)
            {
                int c1 = 2 * cur + 1;
                int c2 = 2 * (cur + 1);
                if (c1 == n - 1)
                {
                    comp = c1;
                }
                else
                {

                    comp = Func_Compare(list[c1], list[c2]) < 0 ? c1 : c2;
                }

                if (Func_Compare(list[cur], list[comp]) > 0)
                {
                    Swap(Q, cur, comp);
                }
                else { break; }

                cur = comp;
            }
        }
        private void Swap(List<T> list, int a, int b)
        {
            T keep = list[a];
            list[a] = list[b];
            list[b] = keep;
        }

        public void Enqueue(T value)
        {
            PushHeap(Q, value);
        }

        public T Dequeue()
        {
            T ret = Q[0];
            PopHeap(Q);
            return ret;
        }

        public T Peek()
        {
            return Q[0];
        }

        public int Count()
        {
            return Q.Count();
        }
        public bool Any()
        {
            return Q.Any();
        }
    }
    public class SegmentTree<T>
    {
        //1-indexed type
        int n;
        T[] Tree;
        Func<T, T, T> f;
        T ex;
        int L;
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public SegmentTree(int size, Func<T, T, T> fun, T exvalue)
        {
            ex = exvalue;
            f = fun;
            n = 1;
            while (n < size) n <<= 1;
            Tree = new T[n << 1];
            L = (n << 1) - 1;
            for (var i = 0; i <= L; i++) Tree[i] = ex;
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public SegmentTree(int size, Func<T, T, T> fun, T exvalue, T[] initial)
        {
            ex = exvalue;
            n = 1; while (n < size) n <<= 1;
            f = fun;
            Tree = new T[n << 1];
            L = (n << 1) - 1;
            for (var i = 0; i <= L; i++) Tree[i] = (n <= i && i <= n + initial.Length - 1) ? initial[i - n] : ex;
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public void Set_All()
        {
            for (var i = n - 1; i >= 1; i--) Tree[i] = f(Tree[i << 1], Tree[(i << 1) | 1]);
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public void Assign(int idx, T nxt) => Tree[idx + n] = nxt;
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public void Update(int idx)
        {
            int now = idx + n;
            while (now > 1)
            {
                now >>= 1;
                Tree[now] = f(Tree[now << 1], Tree[now << 1 | 1]);
            }
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public void Query_Update(int idx, T nxt)
        {
            Assign(idx, nxt); Update(idx);
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public void Query_Update_func(int idx, T y)
        {
            Assign(idx, f(Peek(idx), y)); Update(idx);
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public T Query_Fold(int l, int r)
        {
            int L = n + l;
            int R = n + r;
            T vL = ex, vR = ex;
            while (L < R)
            {
                if (L % 2 == 1)
                {
                    vL = f(vL, Tree[L]);
                    L++;
                }
                if (R % 2 == 1)
                {
                    vR = f(Tree[R - 1], vR);
                    R--;
                }
                L >>= 1;
                R >>= 1;
            }
            return f(vL, vR);
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public T Peek(int idx) => Tree[idx + n];
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public void Display(int len)
        {
            for (var i = 0; i < len; i++) Console.Write($"{Tree[i + n]} ");
            Console.WriteLine();
        }
    }
    public class UnionFind
    {
        private int[] parent;
        private int[] rank;
        private int[] size;
        public UnionFind(int n)
        {
            parent = new int[n];
            rank = new int[n];
            size = new int[n];
            for (var i = 0; i < n; i++)
            {
                parent[i] = i;
                rank[i] = 0;
                size[i] = 1;
            }
        }

        public int Root(int x)
        {
            return parent[x] == x ? x : parent[x] = Root(parent[x]);
        }

        public bool SameRoot(int x, int y)
        {
            return Root(x) == Root(y);
        }

        public void Unite(int x, int y)
        {
            x = Root(x);
            y = Root(y);
            if (x == y) { return; }

            if (rank[x] < rank[y])
            {
                parent[x] = y;
                size[y] += size[x];
                size[x] = 0;
            }
            else
            {
                parent[y] = x;
                if (rank[x] == rank[y]) { rank[x]++; }
                size[x] += size[y];
                size[y] = 0;
            }
        }

        public int SizeOf(int x)
        {
            return size[Root(x)];
        }

    }
    static class Cin
    {
        public static string[] ReadSplit => Console.ReadLine().Split(' ', StringSplitOptions.RemoveEmptyEntries);
        public static int[] ReadSplitInt => ConvertAll(ReadSplit, int.Parse);
        public static long[] ReadSplitLong => ConvertAll(ReadSplit, long.Parse);
        public static double[] ReadSplit_Double => ConvertAll(ReadSplit, double.Parse);
        public static string Str => Console.ReadLine();
        public static int Int => int.Parse(Console.ReadLine());
        public static long Long => long.Parse(Console.ReadLine());
        public static double Double => double.Parse(Console.ReadLine());
        public static T Conv<T>(string input)
        {
            if (typeof(T).Equals(typeof(ModInt)))
            {
                return (T)(dynamic)(long.Parse(input));
            }
            return (T)Convert.ChangeType(input, typeof(T));
        }
        public static void Scanf<T>(out T a) => a = Conv<T>(Console.ReadLine());
        public static void Scanf<T, U>(out T a, out U b)
        { var q = ReadSplit; a = Conv<T>(q[0]); b = Conv<U>(q[1]); }
        public static void Scanf<T, U, V>(out T a, out U b, out V c)
        { var q = ReadSplit; a = Conv<T>(q[0]); b = Conv<U>(q[1]); c = Conv<V>(q[2]); }
        public static void Scanf<T, U, V, W>(out T a, out U b, out V c, out W d)
        { var q = ReadSplit; a = Conv<T>(q[0]); b = Conv<U>(q[1]); c = Conv<V>(q[2]); d = Conv<W>(q[3]); }
        public static void Scanf<T, U, V, W, X>(out T a, out U b, out V c, out W d, out X e)
        { var q = ReadSplit; a = Conv<T>(q[0]); b = Conv<U>(q[1]); c = Conv<V>(q[2]); d = Conv<W>(q[3]); e = Conv<X>(q[4]); }
    }
    static class Cout
    {
        public static void OutL(object s) => Console.WriteLine(s);
        public static void Out_Sep<T>(IEnumerable<T> s) => Console.WriteLine(string.Join(" ", s));
        public static void Out_Sep<T>(IEnumerable<T> s, string sep) => Console.WriteLine(string.Join($"{sep}", s));
        public static void Out_Sep(params object[] s) => Console.WriteLine(string.Join(" ", s));
        public static void Out_One(object s) => Console.Write($"{s} ");
        public static void Out_One(object s, string sep) => Console.Write($"{s}{sep}");
        public static void Endl() => Console.WriteLine();
    }
    public static class Tool
    {
        static public void Initialize<T>(ref T[] array, T initialvalue)
        {
            array = ConvertAll(array, x => initialvalue);
        }
        static public void Swap<T>(ref T a, ref T b)
        {
            T keep = a;
            a = b;
            b = keep;
        }
        static public void Display<T>(T[,] array2d, int n, int m)
        {
            for (var i = 0; i < n; i++)
            {
                for (var j = 0; j < m; j++)
                {
                    Console.Write($"{array2d[i, j]} ");
                }
                Console.WriteLine();
            }
        }
        static public long Gcd(long a, long b)
        {
            if (a == 0 || b == 0) return Max(a, b);
            return a % b == 0 ? b : Gcd(b, a % b);
        }
        static public long LPow(int a, int b) => (long)Pow(a, b);
        static public bool Bit(long x, int dig) => ((1L << dig) & x) != 0;
        static public int Sig(long a) => a == 0 ? 0 : (int)(a / Abs(a));
    }
}