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.Collections;
using System.Collections.Generic;
using System.Diagnostics;
using System.Globalization;
using System.IO;
using System.Text;
using System.Linq;

using E = System.Linq.Enumerable;
using Weight = System.Int64;
using System.Threading;

internal partial class Solver {
    public void Run() {
        var n = ni();
        var k = ni();
        var a = new int[n, n];
        for (int i = 0; i < n; i++) {
            for (int j = 0; j < n; j++) {
                a[i, j] = ni();
            }
        }

        // source -> rows -> a[i,j] -> cols -> sink
        int Row(int row) => row;
        int Col(int col) => col + n;
        var source = 2 * n;
        var sink = source + 1;
        var pd = new PrimalDual(sink + 1);
        for (int i = 0; i < n; i++) {
            pd.AddEdge(source, Row(i), k, 0);
            pd.AddEdge(Col(i), sink, k, 0);
        }

        var inf = n * n * 1000000000L * 10;

        for (int i = 0; i < n; i++) {
            for (int j = 0; j < n; j++) {
                pd.AddEdge(Row(i), Col(j), 1, inf - a[i, j]);
            }
        }

        pd.AddEdge(source, sink, n * k, inf);

        var res = pd.Run(source, sink, n * k);
        long ans = res.TotalFlow * inf - res.TotalCost;
        cout.WriteLine(ans);

        var map = E.Range(0, n).Select(_ => new StringBuilder(new string('.', n))).ToArray();

        foreach (var edge in pd.GetEdges()) {
            if (edge.From < n && n <= edge.To && edge.To < 2 * n && edge.Flow == 1) {
                map[edge.From][edge.To - n] = 'X';
            }
        }

        for (int i = 0; i < n; i++) {
            cout.WriteLine(map[i]);
        }
    }
}

public class MinCostFlowResult {
    public Weight TotalCost;
    public Weight TotalFlow;
}

public class PrimalDual {
    private readonly int N;
    private readonly List<List<Edge>> adjacents;

    private readonly Weight[] _distance;
    private readonly Weight[] _potential;
    private readonly Edge[] _prevEdges;
    private readonly PriorityQueue<Data> _queue;

    public class Edge {
        public int From, To;
        public Weight Flow, Capacity, Cost;
        public Edge Reverse;
        public Edge(int from, int to, Weight capacity, Weight cost, Weight flow = 0) {
            From = from;
            To = to;
            Cost = cost;
            Flow = flow;
            Capacity = capacity;
        }
        public Weight Residue {
            get {
                return Capacity - Flow;
            }
        }
    }

    public PrimalDual(int n) {
        N = n;
        adjacents = new List<List<Edge>>();
        for (int i = 0; i < n; i++) {
            adjacents.Add(new List<Edge>());
        }
        _distance = new Weight[N];
        _potential = new Weight[N];
        _prevEdges = new Edge[N];
        _queue = new PriorityQueue<Data>((d1, d2) => d1.Cost.CompareTo(d2.Cost));
    }

    public IEnumerable<Edge> GetEdges() {
        return adjacents.SelectMany(a => a);
    }

    public void AddEdge(int from, int to, Weight capacity, Weight cost) {
        var e = new Edge(from, to, capacity, cost);
        adjacents[from].Add(e);
    }

    struct Data {
        public int Position;
        public Weight Cost;
    }

    private Edge[] FindShortestPathTree(int source) {
        Array.Clear(_prevEdges, 0, _prevEdges.Length);
        for (int i = 0; i < N; i++) {
            _distance[i] = Weight.MaxValue;
        }
        _distance[source] = 0;
        _queue.Enqueue(new Data { Position = source, Cost = 0 });
        while (_queue.Count > 0) {
            var current = _queue.Dequeue();
            if (current.Cost > _distance[current.Position]) continue; ;
            foreach (var edge in adjacents[current.Position]) {
                if (edge.Residue > 0) {
                    var newCost = _distance[edge.From] + edge.Cost + _potential[edge.From] - _potential[edge.To];
                    if (_distance[edge.To] > newCost) {
                        _distance[edge.To] = newCost;
                        _prevEdges[edge.To] = edge;
                        _queue.Enqueue(new Data { Position = edge.To, Cost = newCost });
                    }
                }
            }
        }
        for (int i = 0; i < N; i++) {
            _potential[i] += _distance[i];
        }

        return _prevEdges;
    }

    public MinCostFlowResult Run(int source, int sink, Weight flow) {
        Weight totalCost = 0;
        Weight totalFlow = 0;

        Array.Clear(_potential, 0, _potential.Length);

        while (totalFlow < flow) {
            var prevEdges = FindShortestPathTree(source);
            if (prevEdges[sink] == null) {
                break;
            }

            long incrementalFlow = flow - totalFlow;
            for (var e = prevEdges[sink]; e != null; e = prevEdges[e.From]) {
                incrementalFlow = Math.Min(incrementalFlow, e.Residue);
            }
            for (var e = prevEdges[sink]; e != null; e = prevEdges[e.From]) {
                if (e.Reverse == null) {
                    var reverseEdge = new Edge(e.To, e.From, 0, -e.Cost) {
                        Reverse = e
                    };
                    e.Reverse = reverseEdge;
                    adjacents[e.To].Add(reverseEdge);
                }
                totalCost += e.Cost * incrementalFlow;
                e.Flow += incrementalFlow;
                e.Reverse.Flow -= incrementalFlow;
            }
            totalFlow += incrementalFlow;
        }
        return new MinCostFlowResult { TotalCost = totalCost, TotalFlow = totalFlow };
    }
}


public class PriorityQueue<T> {
    private T[] _array = new T[128];
    private int _size = 0;
    private readonly Comparison<T> _compare;

    public PriorityQueue() {
        _compare = Comparer<T>.Default.Compare;
    }

    public PriorityQueue(Comparison<T> comp) {
        _compare = comp;
    }

    public int Count { get { return _size; } }

    private void Swap(int a, int b) {
        var t = _array[a];
        _array[a] = _array[b];
        _array[b] = t;
    }

    private void Expand() {
        var newlist = new T[_array.Length << 1];
        Array.Copy(_array, newlist, _array.Length);
        _array = newlist;
    }

    public bool Any() {
        return _size > 0;
    }

    public void Enqueue(T newValue) {
        int pos = ++_size;
        if (_size >= _array.Length) {
            Expand();
        }
        _array[pos] = newValue;
        while (pos > 1) {
            int parent = pos >> 1;
            if (_compare(_array[parent], _array[pos]) > 0) {
                Swap(parent, pos);
                pos = parent;
            } else {
                break;
            }
        }
    }

    public T Peek() {
        return _array[1];
    }

    public T Dequeue() {
        var top = _array[1];
        int pos = 1;
        _array[pos] = _array[_size--];
        while ((pos << 1) <= _size) {
            int left = pos << 1;
            int right = left | 1;
            int next = left;
            if (right <= _size && _compare(_array[left], _array[right]) > 0) {
                next = right;
            }
            if (_compare(_array[pos], _array[next]) > 0) {
                Swap(next, pos);
                pos = next;
            } else {
                break;
            }
        }

        return top;
    }
};

// PREWRITEN CODE BEGINS FROM HERE

static public class StringExtensions {
    static public string JoinToString<T>(this IEnumerable<T> source, string separator = " ") {
        return string.Join(separator, source);
    }
}

internal partial class Solver : Scanner {
    static readonly int? StackSizeInMebiByte = null; //50;
    public static void StartAndJoin(Action action, int maxStackSize) {
        var thread = new Thread(new ThreadStart(action), maxStackSize);
        thread.Start();
        thread.Join();
    }

    public static void Main() {
#if LOCAL
        byte[] inputBuffer = new byte[1000000];
        var inputStream = Console.OpenStandardInput(inputBuffer.Length);
        using (var reader = new StreamReader(inputStream, Console.InputEncoding, false, inputBuffer.Length)) {
            Console.SetIn(reader);
            new Solver(Console.In, Console.Out).Run();
        }
#else
        Console.SetOut(new StreamWriter(Console.OpenStandardOutput()) { AutoFlush = false });
        if (StackSizeInMebiByte.HasValue) {
            StartAndJoin(() => new Solver(Console.In, Console.Out).Run(), StackSizeInMebiByte.Value * 1024 * 1024);
        } else {
            new Solver(Console.In, Console.Out).Run();
        }
        Console.Out.Flush();
#endif
    }

#pragma warning disable IDE0052
    private readonly TextReader cin;
    private readonly TextWriter cout;
    private readonly TextWriter cerr;
#pragma warning restore IDE0052

    public Solver(TextReader reader, TextWriter writer)
        : base(reader) {
        cin = reader;
        cout = writer;
        cerr = Console.Error;
    }

    public Solver(string input, TextWriter writer)
        : this(new StringReader(input), writer) {
    }

#pragma warning disable IDE1006
#pragma warning disable IDE0051
    private int ni() { return NextInt(); }
    private int[] ni(int n) { return NextIntArray(n); }
    private long nl() { return NextLong(); }
    private long[] nl(int n) { return NextLongArray(n); }
    private double nd() { return NextDouble(); }
    private double[] nd(int n) { return NextDoubleArray(n); }
    private string ns() { return Next(); }
    private string[] ns(int n) { return NextArray(n); }
#pragma warning restore IDE1006
#pragma warning restore IDE0051
}

#if DEBUG
internal static class LinqPadExtension {
    public static string TextDump<T>(this T obj) {
        if (obj is IEnumerable) return (obj as IEnumerable).Cast<object>().JoinToString().Dump();
        else return obj.ToString().Dump();
    }
    public static T Dump<T>(this T obj) {
        return LINQPad.Extensions.Dump(obj);
    }
}
#endif

public class Scanner {
    private readonly TextReader Reader;
    private readonly CultureInfo ci = CultureInfo.InvariantCulture;

    private readonly char[] buffer = new char[2 * 1024];
    private int cursor = 0, length = 0;
    private string Token;
    private readonly StringBuilder sb = new StringBuilder(1024);

    public Scanner()
        : this(Console.In) {
    }

    public Scanner(TextReader reader) {
        Reader = reader;
    }

    public int NextInt() { return checked((int)NextLong()); }
    public long NextLong() {
        var s = Next();
        long r = 0;
        int i = 0;
        bool negative = false;
        if (s[i] == '-') {
            negative = true;
            i++;
        }
        for (; i < s.Length; i++) {
            r = r * 10 + (s[i] - '0');
#if DEBUG
            if (!char.IsDigit(s[i])) throw new FormatException();
#endif
        }
        return negative ? -r : r;
    }
    public double NextDouble() { return double.Parse(Next(), ci); }
    public string[] NextArray(int size) {
        string[] array = new string[size];
        for (int i = 0; i < size; i++) {
            array[i] = Next();
        }

        return array;
    }
    public int[] NextIntArray(int size) {
        int[] array = new int[size];
        for (int i = 0; i < size; i++) {
            array[i] = NextInt();
        }

        return array;
    }

    public long[] NextLongArray(int size) {
        long[] array = new long[size];
        for (int i = 0; i < size; i++) {
            array[i] = NextLong();
        }

        return array;
    }

    public double[] NextDoubleArray(int size) {
        double[] array = new double[size];
        for (int i = 0; i < size; i++) {
            array[i] = NextDouble();
        }

        return array;
    }

    public string Next() {
        if (Token == null) {
            if (!StockToken()) {
                throw new Exception();
            }
        }
        var token = Token;
        Token = null;
        return token;
    }

    public bool HasNext() {
        if (Token != null) {
            return true;
        }

        return StockToken();
    }

    private bool StockToken() {
        while (true) {
            sb.Clear();
            while (true) {
                if (cursor >= length) {
                    cursor = 0;
                    if ((length = Reader.Read(buffer, 0, buffer.Length)) <= 0) {
                        break;
                    }
                }
                var c = buffer[cursor++];
                if (33 <= c && c <= 126) {
                    sb.Append(c);
                } else {
                    if (sb.Length > 0) break;
                }
            }

            if (sb.Length > 0) {
                Token = sb.ToString();
                return true;
            }

            return false;
        }
    }
}