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 PyPy3 (7.3.0) (judge verdict: Accepted)

import heapq
# numeric_limitsわかんね
NUMERIC_LIMITS = 10 ** 18

class mcf_graph:
  def __init__ (s, n):
    s._n = n
    s.g = [[] for _ in range(n)]
    s.pos = []

  def add_edge(s, frm, to, cap, cost):
    m = len(s.pos)
    s.pos.append([frm, len(s.g[frm])])
    s.g[frm].append(s._edge(to, len(s.g[to]), cap, cost))
    s.g[to].append(s._edge(frm, len(s.g[frm]) - 1, 0, -cost))
    return m

  class edge:
    __slots__ = "frm", "to", "cap", "flow", "cost"
    def __init__(s, frm, to, cap, flow, cost):
      s.frm, s.to = frm, to
      s.cap, s.flow, s.cost = cap, flow, cost

  def get_edge(s, i):
    _e = s.g[s.pos[i][0]][s.pos[i][1]]
    _re = s.g[_e.to][_e.rev]
    return s.edge(s.pos[i][0], _e.to, _e.cap + _re.cap, _re.cap, _e.cost)

  def edges(s):
    for i in range(len(s.pos)):
      yield s.get_edge(i)

  def flow(self, s, t, flow_limit = NUMERIC_LIMITS):
    return self.slope(s, t, flow_limit)[-1]

  def dual_ref(self, s, t):
    #priority_queueの代わり
    push = heapq.heappush
    pop = heapq.heappop

    dist = [NUMERIC_LIMITS] * self._n
    self.pv = [-1] * self._n
    self.pe = [-1] * self._n
    vis = [False] * self._n
    que = []
    dist[s] = 0
    push(que, (0, s))
    while que:
      k, v = pop(que)
      if vis[v]: continue
      vis[v] = True
      if v == t: break
      for i in range(len(self.g[v])):
        e = self.g[v][i]
        if vis[e.to] or not e.cap: continue
        cost = e.cost - self.dual[e.to] + self.dual[v]
        if dist[e.to] - dist[v] > cost:
          dist[e.to] = dist[v] + cost
          self.pv[e.to] = v
          self.pe[e.to] = i
          push(que,(dist[e.to], e.to))
    if not vis[t]: return False
    for v in range(self._n):
      if not vis[v]: continue
      self.dual[v] -= dist[t] - dist[v]
    return True

  def slope(self, s, t, flow_limit = NUMERIC_LIMITS):
    self.dual = [0] * self._n
  
    flow = 0
    cost, prev_cost = 0, -1
    result = [(flow,cost)]
    while flow < flow_limit:
      if not self.dual_ref(s, t): break
      c = flow_limit - flow
      v = t
      while v != s:
        c = min(c, self.g[self.pv[v]][self.pe[v]].cap)
        v = self.pv[v]
      v = t
      while v != s:
        e = self.g[self.pv[v]][self.pe[v]]
        e.cap -= c
        self.g[v][e.rev].cap += c
        v = self.pv[v]
      d = -self.dual[s]
      flow += c
      cost += c * d
      if prev_cost == d:
        result.pop()
      result.append((flow, cost))
      prev_cost = cost
    return result

  class _edge:
    __slots__ = "to", "rev", "cap", "cost"
    def __init__(s, to, rev, cap, cost):
      s.to, s.rev = to, rev
      s.cap, s.cost = cap, cost

N, K = list(map(int, input().split()))
A = [tuple(map(int, input().split())) for _ in range(N)] 
BIG = 10 ** 9

g = mcf_graph(2 * N + 2)
s = 2 * N
t = 2 * N + 1

g.add_edge(s, t, N * K, BIG)

#   k,0|1,A[i][j]|k,0
#      列0 ー 行0
#   ／　　 ×     ＼
# s ー 列1 米 行1 ー t
#   ＼  　 ×     ／
#      列i ー 行j
for i in range(N):
  g.add_edge(s, i, K, 0)
  g.add_edge(N + i, t, K, 0)
for i in range(N):
  for j in range(N):
    g.add_edge(i, N + j, 1, BIG - A[i][j])

result = g.flow(s, t, N * K)
print(N * K * BIG - result[1])

grid = [["."] * N for _ in range(N)]
edges = g.edges()
for e in edges:
  if e.frm == s or e.to == t or e.flow == 0: continue
  grid[e.frm][e.to - N] = "X"

for g in grid:
  print("".join(g))