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.Generic;
using System.Linq;
using System.IO;
using System.Threading;
using System.Runtime.CompilerServices;
using System.Text;
using System.Diagnostics;
using System.Numerics;
using static System.Console;
using static System.Convert;
using static System.Math;
using static Template;
using SC = Scanner;

public partial class Solver
{
    public void Solve()
    {
        int N, K;
        sc.Make(out N, out K);
        var A = sc.ArrInt2D(N);
        var G = new PrimalDual(2 * N + 2);
        int s = 2 * N, t = 2 * N + 1;
        G.AddEdge(s, t, N * K, (int)1e9);
        for (int i = 0; i < N; i++)
        {
            G.AddEdge(s, i, K, 0);
            G.AddEdge(N + i, t, K, 0);
        }
        for (int i = 0; i < N; i++)
        {
            for (int j = 0; j < N; j++)
            {
                G.AddEdge(i, N + j, 1, (int)1e9 - A[i][j]);
            }
        }
        var res = G.flow(s, t, N * K);
        Console.WriteLine(N * K * (long)1e9 - res.Item2);
        var grd = Create(N, () => Create(N, () => '.'));
        for (int i = 0; i < G.EdgeCount; i += 2)
        {
            var e = G.GetEdge(i);
            if (e.from == s || e.to == t || e.flow == 0) continue;
            grd[e.from][e.to - N] = 'X';
        }
        Console.WriteLine(grd.Select(v => new string(v)).Concat("\n"));
    }
}
class PrimalDual
{
    const long Inf = long.MaxValue / 2;
    List<_Edge> edges;
    public int EdgeCount => edges.Count;
    List<int>[] G;
    public PrimalDual(int N)
    {
        G = Create(N, () => new List<int>());
        edges = new List<_Edge>();
    }
    public void AddEdge(int from, int to, long cap, long cost)
    {
        int edgeIndex = edges.Count;
        G[from].Add(edgeIndex);
        edges.Add(new _Edge(to, edgeIndex, cap, cost));
        G[to].Add(edgeIndex | 1);
        edges.Add(new _Edge(from, edgeIndex | 1, 0, -cost));
    }
    public Edge GetEdge(int i) => new Edge { from = edges[i ^ 1].to, to = edges[i].to, cap = edges[i].cap + edges[i ^ 1].cap, flow = edges[i ^ 1].cap, cost = edges[i].cost };
    //flow,cost
    public (long, long) flow(int s, int t, long f = Inf) => Slope(s, t, f).Last();
    public List<(long, long)> Slope(int s, int t, long f)
    {
        int N = G.Length;
        var dual = new long[N]; var dist = new long[N];//cost
        var pv = new int[N]; var pe = new int[N];//最短経路木のようなもの
        var used = new bool[N];
        Func<bool> SP = null;
        SP = () =>
        {
            for (int i = 0; i < dist.Length; i++)
            {
                dist[i] = Inf;
                pv[i] = pe[i] = -1;
                used[i] = false;
            }
            var pq = new PriorityQueue<(long dist, int to)>((a, b) => a.dist.CompareTo(b.dist));
            dist[s] = 0;
            pq.Push((0, s));
            while (pq.Any())
            {
                var v = pq.Pop().to;
                if (used[v]) continue;
                used[v] = true;
                if (v == t) break;
                for (int i = 0; i < G[v].Count; i++)
                {
                    var e = edges[G[v][i]];
                    if (used[e.to] || e.cap == 0) continue;
                    var cost = e.cost - dual[e.to] + dual[v];
                    if (dist[e.to] - dist[v] > cost)
                    {
                        dist[e.to] = dist[v] + cost;
                        pv[e.to] = v;
                        pe[e.to] = e.idx;
                        pq.Push((dist[e.to], e.to));
                    }
                }
            }
            if (!used[t]) return false;
            for (int i = 0; i < dist.Length; i++)
            {
                if (!used[i]) continue;
                dual[i] -= dist[t] - dist[i];//ポテンシャル
            }
            return true;
        };
        long flow = 0, cost = 0, prev = -1;
        var res = new List<(long, long)>();
        res.Add((flow, cost));
        while (flow < f)
        {
            if (!SP()) break;
            var c = f - flow;
            for (var v = t; v != s; v = pv[v]) c = Min(c, edges[pe[v]].cap);
            for (int v = t; v != s; v = pv[v])
            {
                edges[pe[v]].cap -= c;
                edges[pe[v] ^ 1].cap += c;
            }
            var d = -dual[s];
            flow += c;
            cost += c * d;
            if (prev == d) res.RemoveAt(res.Count - 1);
            res.Add((flow, cost));
            prev = d;
        }
        return res;
    }
    public struct Edge { public int from, to; public long cap, cost, flow; }
    public class _Edge
    {
        public int to, idx;
        public long cap, cost;
        public _Edge(int t, int i, long ca, long co)
        {
            to = t; idx = i; cap = ca; cost = co;
        }
    }
}

public class PriorityQueue<T>
{
    private List<T> data = new List<T>();
    private Comparison<T> cmp;
    public int Count { get { return data.Count; } }
    public T Top { get { return data[0]; } }
    public PriorityQueue() { cmp = cmp ?? Comparer<T>.Default.Compare; }

    public PriorityQueue(Comparison<T> comparison) { cmp = comparison; }

    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    private int Parent(int i)
        => (i - 1) >> 1;
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    private int Left(int i)
        => (i << 1) + 1;
    public T Push(T val)
    {
        int i = data.Count;
        data.Add(val);
        while (i > 0)
        {
            int p = Parent(i);
            if (cmp(data[p], val) <= 0)
                break;
            data[i] = data[p];
            i = p;
        }
        data[i] = val;
        return val;
    }
    public T Pop()
    {
        var ret = data[0];
        var p = 0;
        var x = data[data.Count - 1];
        while (Left(p) < data.Count - 1)
        {
            var l = Left(p);
            if (l < data.Count - 2 && cmp(data[l + 1], data[l]) < 0) l++;
            if (cmp(data[l], x) >= 0)
                break;
            data[p] = data[l];
            p = l;
        }
        data[p] = x;
        data.RemoveAt(data.Count - 1);
        return ret;
    }
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public bool Any() => data.Count > 0;
}

#region Template
public partial class Solver
{
    public SC sc = new SC();
    static void Main(string[] args)
    {
        Console.SetOut(new StreamWriter(Console.OpenStandardOutput()) { AutoFlush = false });
        var sol = new Solver();
        int testcase = 1;
        //testcase = sol.sc.Int;
        //var th = new Thread(sol.Solve, 1 << 26);th.Start();th.Join();
        while (testcase-- > 0)
            sol.Solve();
        Console.Out.Flush();
    }
}
public static class Template
{
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static bool chmin<T>(ref T a, T b) where T : IComparable<T> { if (a.CompareTo(b) > 0) { a = b; return true; } return false; }
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static bool chmax<T>(ref T a, T b) where T : IComparable<T> { if (a.CompareTo(b) < 0) { a = b; return true; } return false; }
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static void swap<T>(ref T a, ref T b) { var t = b; b = a; a = t; }
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static void swap<T>(this IList<T> A, int i, int j) { var t = A[i]; A[i] = A[j]; A[j] = t; }
    public static T[] Create<T>(int n, Func<T> f) { var rt = new T[n]; for (var i = 0; i < rt.Length; ++i) rt[i] = f(); return rt; }
    public static T[] Create<T>(int n, Func<int, T> f) { var rt = new T[n]; for (var i = 0; i < rt.Length; ++i) rt[i] = f(i); return rt; }
    public static void Out<T>(this IList<T> A, out T a) => a = A[0];
    public static void Out<T>(this IList<T> A, out T a, out T b) { a = A[0]; b = A[1]; }
    public static void Out<T>(this IList<T> A, out T a, out T b, out T c) { A.Out(out a, out b); c = A[2]; }
    public static void Out<T>(this IList<T> A, out T a, out T b, out T c, out T d) { A.Out(out a, out b, out c); d = A[3]; }
    public static string Concat<T>(this IEnumerable<T> A, string sp) => string.Join(sp, A);
    public static char ToChar(this int s, char begin = '0') => (char)(s + begin);
    public static IEnumerable<T> Shuffle<T>(this IEnumerable<T> A) => A.OrderBy(v => Guid.NewGuid());
    public static int CompareTo<T>(this T[] A, T[] B, Comparison<T> cmp = null) { cmp = cmp ?? Comparer<T>.Default.Compare; for (var i = 0; i < Min(A.Length, B.Length); i++) { int c = cmp(A[i], B[i]); if (c > 0) return 1; else if (c < 0) return -1; } if (A.Length == B.Length) return 0; if (A.Length > B.Length) return 1; else return -1; }
    public static string ToStr<T>(this T[][] A) => A.Select(a => a.Concat(" ")).Concat("\n");
    public static int ArgMax<T>(this IList<T> A, Comparison<T> cmp = null) { cmp = cmp ?? Comparer<T>.Default.Compare; T max = A[0]; int rt = 0; for (int i = 1; i < A.Count; i++) if (cmp(max, A[i]) < 0) { max = A[i]; rt = i; } return rt; }
    public static T PopBack<T>(this List<T> A) { var v = A[A.Count - 1]; A.RemoveAt(A.Count - 1); return v; }
    public static void Fail<T>(T s) { Console.WriteLine(s); Console.Out.Close(); Environment.Exit(0); }
}
public class Scanner
{
    public string Str => Console.ReadLine().Trim();
    public int Int => int.Parse(Str);
    public long Long => long.Parse(Str);
    public double Double => double.Parse(Str);
    public int[] ArrInt => Str.Split(' ').Select(int.Parse).ToArray();
    public long[] ArrLong => Str.Split(' ').Select(long.Parse).ToArray();
    public char[][] Grid(int n) => Create(n, () => Str.ToCharArray());
    public int[] ArrInt1D(int n) => Create(n, () => Int);
    public long[] ArrLong1D(int n) => Create(n, () => Long);
    public int[][] ArrInt2D(int n) => Create(n, () => ArrInt);
    public long[][] ArrLong2D(int n) => Create(n, () => ArrLong);
    private Queue<string> q = new Queue<string>();
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public T Next<T>() { if (q.Count == 0) foreach (var item in Str.Split(' ')) q.Enqueue(item); return (T)Convert.ChangeType(q.Dequeue(), typeof(T)); }
    public void Make<T1>(out T1 v1) => v1 = Next<T1>();
    public void Make<T1, T2>(out T1 v1, out T2 v2) { v1 = Next<T1>(); v2 = Next<T2>(); }
    public void Make<T1, T2, T3>(out T1 v1, out T2 v2, out T3 v3) { Make(out v1, out v2); v3 = Next<T3>(); }
    public void Make<T1, T2, T3, T4>(out T1 v1, out T2 v2, out T3 v3, out T4 v4) { Make(out v1, out v2, out v3); v4 = Next<T4>(); }
    public void Make<T1, T2, T3, T4, T5>(out T1 v1, out T2 v2, out T3 v3, out T4 v4, out T5 v5) { Make(out v1, out v2, out v3, out v4); v5 = Next<T5>(); }
    public void Make<T1, T2, T3, T4, T5, T6>(out T1 v1, out T2 v2, out T3 v3, out T4 v4, out T5 v5, out T6 v6) { Make(out v1, out v2, out v3, out v4, out v5); v6 = Next<T6>(); }
    public void Make<T1, T2, T3, T4, T5, T6, T7>(out T1 v1, out T2 v2, out T3 v3, out T4 v4, out T5 v5, out T6 v6, out T7 v7) { Make(out v1, out v2, out v3, out v4, out v5, out v6); v7 = Next<T7>(); }
    //public (T1, T2) Make<T1, T2>() { Make(out T1 v1, out T2 v2); return (v1, v2); }
    //public (T1, T2, T3) Make<T1, T2, T3>() { Make(out T1 v1, out T2 v2, out T3 v3); return (v1, v2, v3); }
    //public (T1, T2, T3, T4) Make<T1, T2, T3, T4>() { Make(out T1 v1, out T2 v2, out T3 v3, out T4 v4); return (v1, v2, v3, v4); }
}

#endregion