Problem:
Score : 100 points

Problem Statement

You are given a grid of N rows and M columns. The square at the i-th row and j-th column will be denoted as (i,j).
A nonnegative integer A_{i,j} is written for each square (i,j).

You choose some of the squares so that each row and column contains at most K chosen squares.
Under this constraint, calculate the maximum value of the sum of the integers written on the chosen squares.
Additionally, calculate a way to choose squares that acheives the maximum.

Constraints

1 \leq N \leq 50

1 \leq K \leq N

0 \leq A_{i,j} \leq 10^9

All values in Input are integer.

Input

Input is given from Standard Input in the following format:

N K
A_{1,1} A_{1,2} \cdots A_{1,N}
A_{2,1} A_{2,2} \cdots A_{2,N}
\vdots
A_{N,1} A_{N,2} \cdots A_{N,N}

Output

On the first line, print the maximum value of the sum of the integers written on the chosen squares.

On the next N lines, print a way that achieves the maximum.

Precisely, output the strings t_1,t_2,\cdots,t_N, that satisfies t_{i,j}=X if you choose (i,j) and t_{i,j}=. otherwise.

You may print any way to choose squares that maximizes the sum.

Sample Input 1

3 1
5 3 2
1 4 8
7 6 9

Sample Output 1

19
X..
..X
.X.

Sample Input 2

3 2
10 10 1
10 10 1
1 1 10

Sample Output 2

50
XX.
XX.
..X

Solution in C# (.NET Core 3.1.201) (judge verdict: Accepted)

using System;
using System.Linq;
using CompLib.Util;
using System.Threading;
using CompLib.Graph;
using System.Collections.Generic;

public class Program
{

    public void Solve()
    {
        var sc = new Scanner();
        int n = sc.NextInt();
        int k = sc.NextInt();
        int[][] a = new int[n][];
        for (int i = 0; i < n; i++)
        {
            a[i] = sc.IntArray();
        }

        // 0スタート
        // 1~n^2 各マス
        // n^2+1~n^2+n 縦
        // 横
        // ゴール


        const long inf = 1000000000;
        var mcf = new MinCostFlow(1 + 2 * n + 1);
        int s = 2 * n;
        int g = 1 + 2 * n;
        mcf.AddEdge(s, g, n * k, inf);
        for (int i = 0; i < n; i++)
        {
            mcf.AddEdge(s, i, k, 0);
            mcf.AddEdge(n + i, g, k, 0);
        }
        var ls = new List<int>();
        for (int i = 0; i < n; i++)
        {
            for (int j = 0; j < n; j++)
            {
                ls.Add(mcf.AddEdge(i, n + j, 1, inf - a[i][j]));
            }
        }

        var res = mcf.Flow(s, g, n * k);
        long max = (long)n * k * inf - res.Cost;
        Console.WriteLine(max);
        char[][] ans = new char[n][];
        for (int i = 0; i < n; i++)
        {
            ans[i] = new char[n];
            for (int j = 0; j < n; j++)
            {
                ans[i][j] = '.';
            }
        }
        foreach (var i in ls)
        {
            var e = mcf.GetEdge(i);
            if (e.Flow > 0)
            {
                ans[e.From][e.To - n] = 'X';
            }
        }
        for (int i = 0; i < n; i++)
        {
            Console.WriteLine(new string(ans[i]));
        }
    }

    public static void Main(string[] args) => new Program().Solve();
    // public static void Main(string[] args) => new Thread(new Program().Solve, 1 << 27).Start();
}

// https://bitbucket.org/camypaper/complib
namespace CompLib.Collections
{
    using System;
    using System.Collections.Generic;

    #region PriorityQueue

    /// <summary>
    /// 指定した型のインスタンスを最も価値が低い順に取り出すことが可能な可変サイズのコレクションを表します．
    /// </summary>
    /// <typeparam name="T">優先度付きキュー内の要素の型を指定します．</typeparam>
    /// <remarks>内部的にはバイナリヒープによって実装されています．</remarks>
    public class PriorityQueue<T>
    {
        readonly List<T> heap = new List<T>();
        readonly Comparison<T> cmp;

        /// <summary>
        /// デフォルトの比較子を使用してインスタンスを初期化します．
        /// </summary>
        /// <remarks>この操作は O(1) で実行されます．</remarks>
        public PriorityQueue()
        {
            cmp = Comparer<T>.Default.Compare;
        }

        /// <summary>
        /// デリゲートで表されるような比較関数を使用してインスタンスを初期化します．
        /// </summary>
        /// <param name="comparison"></param>
        /// <remarks>この操作は O(1) で実行されます．</remarks>
        public PriorityQueue(Comparison<T> comparison)
        {
            cmp = comparison;
        }

        /// <summary>
        /// 指定された比較子を使用してインスタンスを初期化します．
        /// </summary>
        /// <param name="comparer"></param>
        /// <remarks>この操作は O(1) で実行されます．</remarks>
        public PriorityQueue(IComparer<T> comparer)
        {
            cmp = comparer.Compare;
        }

        /// <summary>
        /// 優先度付きキューに要素を追加します．
        /// </summary>
        /// <param name="item">優先度付きキューに追加される要素</param>
        /// <remarks>最悪計算量 O(log N) で実行されます．</remarks>
        public void Enqueue(T item)
        {
            var pos = heap.Count;
            heap.Add(item);
            while (pos > 0)
            {
                var par = (pos - 1) / 2;
                if (cmp(heap[par], item) <= 0)
                    break;
                heap[pos] = heap[par];
                pos = par;
            }

            heap[pos] = item;
        }

        /// <summary>
        /// 優先度付きキューから最も価値が低い要素を削除し，返します．
        /// </summary>
        /// <returns>優先度付きキューから削除された要素．</returns>
        /// <remarks>最悪計算量 O(log N) で実行されます．</remarks>
        public T Dequeue()
        {
            var ret = heap[0];
            var pos = 0;
            var x = heap[heap.Count - 1];

            while (pos * 2 + 1 < heap.Count - 1)
            {
                var lch = pos * 2 + 1;
                var rch = pos * 2 + 2;
                if (rch < heap.Count - 1 && cmp(heap[rch], heap[lch]) < 0) lch = rch;
                if (cmp(heap[lch], x) >= 0)
                    break;
                heap[pos] = heap[lch];
                pos = lch;
            }

            heap[pos] = x;
            heap.RemoveAt(heap.Count - 1);
            return ret;
        }

        /// <summary>
        ///  優先度付きキューに含まれる最も価値が低い要素を削除せずに返します．
        /// </summary>
        /// <returns>優先度付きキューに含まれる最も価値が低い要素．</returns>
        /// <remarks>この操作は O(1) で実行されます．</remarks>
        public T Peek()
        {
            return heap[0];
        }

        /// <summary>
        /// 優先度付きキュー内の要素の数を取得します．
        /// </summary>
        /// <returns>優先度付キュー内にある要素の数</returns>
        /// <remarks>最悪計算量 O(1) で実行されます．</remarks>
        public int Count
        {
            get { return heap.Count; }
        }

        /// <summary>
        /// 優先度付きキュー内に要素が存在するかどうかを O(1) で判定します．
        /// </summary>
        /// <returns>優先度付キュー内にある要素が存在するならば true，そうでなければ　false．</returns>
        /// <remarks>この操作は O(1) で実行されます．</remarks>
        public bool Any()
        {
            return heap.Count > 0;
        }

        /// <summary>
        /// 優先度付きキューに含まれる要素を昇順に並べて返します．
        /// </summary>
        /// <remarks>この操作は計算量 O(N log N)で実行されます．</remarks>
        public T[] Items
        {
            get
            {
                var ret = heap.ToArray();
                Array.Sort(ret, cmp);
                return ret;
            }
        }
    }

    #endregion
}

namespace CompLib.Graph
{
    using CompLib.Collections;
    using Microsoft.VisualBasic;
    using System;
    using System.Collections.Generic;
    using System.Diagnostics;
    using Cap = System.Int32;
    using Cost = System.Int64;
    class MinCostFlow
    {
        private readonly int N;
        private readonly List<PairII> Pos;
        private readonly List<InternalEdge>[] G;

        public MinCostFlow(int n)
        {
            N = n;
            Pos = new List<PairII>();
            G = new List<InternalEdge>[N];
            for (int i = 0; i < N; i++)
            {
                G[i] = new List<InternalEdge>();
            }
        }

        /// <summary>
        /// fromからtoへ最大容量cap, コストcostの辺を追加する 何番目に追加された辺かを返す
        /// </summary>
        /// <remarks>O(1)</remarks>
        /// <param name="from"></param>
        /// <param name="to"></param>
        /// <param name="cap"></param>
        /// <param name="cost"></param>
        /// <returns></returns>
        public int AddEdge(int from, int to, Cap cap, Cost cost)
        {
            Debug.Assert(0 <= from && from < N);
            Debug.Assert(0 <= to && to < N);
            int m = Pos.Count;
            Pos.Add(new PairII(from, G[from].Count));
            int fromID = G[from].Count;
            int toID = G[to].Count;
            if (from == to) toID++;
            G[from].Add(new InternalEdge(to, toID, cap, cost));
            G[to].Add(new InternalEdge(from, fromID, 0, -cost));
            return m;
        }

        /// <summary>
        /// i番目に追加された辺の状態を返します
        /// </summary>
        /// <remarks>O(1)</remarks>
        /// <param name="i"></param>
        /// <returns></returns>
        public Edge GetEdge(int i)
        {
            int m = Pos.Count;
            Debug.Assert(0 <= i && i < m);
            var e = G[Pos[i].From][Pos[i].Index];
            var re = G[e.To][e.Rev];
            return new Edge(Pos[i].From, e.To, e.Cap + re.Cap, re.Cap, e.Cost);
        }

        /// <summary>
        /// 現在の辺の状態を返します
        /// </summary>
        /// <remarks>O(m)</remarks>
        /// <returns></returns>
        public Edge[] Edges()
        {
            int m = Pos.Count;
            Edge[] result = new Edge[m];
            for (int i = 0; i < m; i++)
            {
                result[i] = GetEdge(i);
            }
            return result;
        }

        public PairCapCost Flow(int s, int t)
        {
            return Flow(s, t, Cap.MaxValue);
        }

        public PairCapCost Flow(int s, int t, Cap flowLimit)
        {
            var ls = Slope(s, t, flowLimit);
            return ls[ls.Count - 1];
        }

        public List<PairCapCost> Slope(int s, int t)
        {
            return Slope(s, t, Cap.MaxValue);
        }

        public List<PairCapCost> Slope(int s, int t, Cap flowLimit)
        {
            Debug.Assert(0 <= s && s < N);
            Debug.Assert(0 <= t && t < N);
            Debug.Assert(s != t);

            Cost[] dual = new Cost[N];
            Cost[] dist = new Cost[N];
            int[] pv = new int[N];
            int[] pe = new int[N];
            bool[] vis = new bool[N];

            bool dualRef()
            {
                for (int i = 0; i < N; i++)
                {
                    dist[i] = Cost.MaxValue;
                    pv[i] = -1;
                    pe[i] = -1;
                    vis[i] = false;
                }

                // C++のstd::priority_queueは降順
                // これは昇順
                PriorityQueue<Q> que = new PriorityQueue<Q>((l, r) => l.Key.CompareTo(r.Key));

                dist[s] = 0;
                que.Enqueue(new Q(0, s));
                while (que.Any())
                {
                    int v = que.Dequeue().To;
                    if (vis[v]) continue;
                    vis[v] = true;
                    if (v == t) break;
                    for (int i = 0; i < G[v].Count; i++)
                    {
                        var e = G[v][i];
                        if (vis[e.To] || e.Cap == 0) continue;
                        Cost cost2 = e.Cost - dual[e.To] + dual[v];
                        if (dist[e.To] - dist[v] > cost2)
                        {
                            dist[e.To] = dist[v] + cost2;
                            pv[e.To] = v;
                            pe[e.To] = i;
                            que.Enqueue(new Q(dist[e.To], e.To));
                        }
                    }
                }

                if (!vis[t])
                {
                    return false;
                }

                for (int v = 0; v < N; v++)
                {
                    if (!vis[v]) continue;
                    dual[v] -= dist[t] - dist[v];
                }

                return true;
            }

            Cap flow = 0;
            Cost cost = 0;
            Cost prevCostPerFlow = -1;
            List<PairCapCost> result = new List<PairCapCost>();
            result.Add(new PairCapCost(flow, cost));
            while (flow < flowLimit)
            {
                if (!dualRef()) break;
                Cap c = flowLimit - flow;
                for (int v = t; v != s; v = pv[v])
                {
                    c = Math.Min(c, G[pv[v]][pe[v]].Cap);
                }

                for (int v = t; v != s; v = pv[v])
                {
                    var e = G[pv[v]][pe[v]];
                    e.Cap -= c;
                    var rev = G[v][e.Rev];
                    rev.Cap += c;
                    G[pv[v]][pe[v]] = e;
                    G[v][e.Rev] = rev;
                }
                Cost d = -dual[s];
                flow += c;
                cost += c * d;
                if (prevCostPerFlow == d)
                {
                    result.RemoveAt(result.Count - 1);
                }
                result.Add(new PairCapCost(flow, cost));
                prevCostPerFlow = d;
            }
            return result;
        }

        private struct PairII
        {
            public int From, Index;
            public PairII(int from, int index)
            {
                From = from;
                Index = index;
            }
        }

        private struct InternalEdge
        {
            public int To, Rev;
            public Cap Cap;
            public Cost Cost;
            public InternalEdge(int to, int rev, Cap cap, Cost cost)
            {
                To = to;
                Rev = rev;
                Cap = cap;
                Cost = cost;
            }
        }

        private struct Q
        {
            public Cost Key;
            public int To;
            public Q(Cost key, int to)
            {
                Key = key;
                To = to;
            }
        }
    }

    public struct Edge
    {
        public readonly int From, To;
        public readonly Cap Cap, Flow;
        public readonly Cost Cost;
        public Edge(int from, int to, Cap cap, Cap flow, Cost cost)
        {
            From = from;
            To = to;
            Cap = cap;
            Flow = flow;
            Cost = cost;
        }
    }

    public struct PairCapCost
    {
        public Cap Cap;
        public Cost Cost;
        public PairCapCost(Cap cap, Cost cost)
        {
            Cap = cap;
            Cost = cost;
        }
    }
}


namespace CompLib.Util
{
    using System;
    using System.Linq;

    class Scanner
    {
        private string[] _line;
        private int _index;
        private const char Separator = ' ';

        public Scanner()
        {
            _line = new string[0];
            _index = 0;
        }

        public string Next()
        {
            if (_index >= _line.Length)
            {
                string s;
                do
                {
                    s = Console.ReadLine();
                } while (s.Length == 0);

                _line = s.Split(Separator);
                _index = 0;
            }

            return _line[_index++];
        }

        public string ReadLine()
        {
            _index = _line.Length;
            return Console.ReadLine();
        }

        public int NextInt() => int.Parse(Next());
        public long NextLong() => long.Parse(Next());
        public double NextDouble() => double.Parse(Next());
        public decimal NextDecimal() => decimal.Parse(Next());
        public char NextChar() => Next()[0];
        public char[] NextCharArray() => Next().ToCharArray();

        public string[] Array()
        {
            string s = Console.ReadLine();
            _line = s.Length == 0 ? new string[0] : s.Split(Separator);
            _index = _line.Length;
            return _line;
        }

        public int[] IntArray() => Array().Select(int.Parse).ToArray();
        public long[] LongArray() => Array().Select(long.Parse).ToArray();
        public double[] DoubleArray() => Array().Select(double.Parse).ToArray();
        public decimal[] DecimalArray() => Array().Select(decimal.Parse).ToArray();
    }
}