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) .
Some of the squares contain an object. All the remaining squares are empty.
The state of the grid is represented by strings S_1,S_2,\cdots,S_N . The square (i,j) contains an object if S_{i,j}=  # and is empty if S_{i,j}=  . .  Consider placing 1 \times 2 tiles on the grid. Tiles can be placed vertically or horizontally to cover two adjacent empty squares.
Tiles must not stick out of the grid, and no two different tiles may intersect. Tiles cannot occupy the square with an object.  Calculate the maximum number of tiles that can be placed and any configulation that acheives the maximum.      Constraints  1 \leq N \leq 100  1 \leq M \leq 100  S_i is a string with length M consists of # and . .         Input Input is given from Standard Input in the following format:  N  M  S_1  S_2  \vdots  S_N       Output On the first line, print the maximum number of tiles that can be placed.  On the next N lines, print a configulation that achieves the maximum.
Precisely, output the strings t_1,t_2,\cdots,t_N constructed by the following way.   t_i is initialized to S_i .  For each (i,j) , if there is a tile that occupies (i,j) and (i+1,j) , change t_{i,j} := v , t_{i+1,j} := ^ .  For each (i,j) , if there is a tile that occupies (i,j) and (i,j+1) , change t_{i,j} := > , t_{i,j+1} := < .   See samples for further information.  You may print any configulation that maximizes the number of tiles.        Sample Input 1 3 3
#..
..#
...      Sample Output 1 3
#><
vv#
^^.  The following output is also treated as a correct answer.  3
#><
v.#
^><

Solution (Python):
import sys
from collections import deque
from copy import deepcopy
sys.setrecursionlimit(10**7)
def MI(): return map(int,sys.stdin.readline().rstrip().split())
def LS2(): return list(sys.stdin.readline().rstrip())  


class Dinic:
    def __init__(self, N):
        self.N = N
        self.G = [[] for _ in range(N)]

    def add_edge(self, fr, to, cap):
        forward = [to, cap, None]
        forward[2] = backward = [fr, 0, forward]
        self.G[fr].append(forward)
        self.G[to].append(backward)

    def add_multi_edge(self, v1, v2, cap1, cap2):
        edge1 = [v2, cap1, None]
        edge1[2] = edge2 = [v1, cap2, edge1]
        self.G[v1].append(edge1)
        self.G[v2].append(edge2)

    def bfs(self, s, t):
        self.level = level = [None]*self.N
        deq = deque([s])
        level[s] = 0
        G = self.G
        while deq:
            v = deq.popleft()
            lv = level[v] + 1
            for w, cap, _ in G[v]:
                if cap and level[w] is None:
                    level[w] = lv
                    deq.append(w)
        return level[t] is not None

    def dfs(self, v, t, f):
        if v == t:
            return f
        level = self.level
        for e in self.it[v]:
            w, cap, rev = e
            if cap and level[v] < level[w]:
                d = self.dfs(w, t, min(f, cap))
                if d:
                    e[1] -= d
                    rev[1] += d
                    return d
        return 0

    def flow(self, s, t):
        flow = 0
        INF = 10**9 + 7
        G = self.G
        while self.bfs(s, t):
            *self.it, = map(iter, self.G)
            f = INF
            while f:
                f = self.dfs(s, t, INF)
                flow += f
        return flow


N,M = MI()
S = [LS2() for _ in range(N)]

D = Dinic(N*M+2)
s,t = N*M,N*M+1
for i in range(N):
    for j in range(M):
        if (i+j) % 2 == 0:
            D.add_edge(s,M*i+j,1)
        else:
            D.add_edge(M*i+j,t,1)

for i in range(N-1):
    for j in range(M):
        if S[i][j] == S[i+1][j] == '.':
            a,b = M*i+j,M*(i+1)+j
            if (i+j) % 2 == 1:
                a,b = b,a
            D.add_edge(a,b,1)

for i in range(N):
    for j in range(M-1):
        if S[i][j] == S[i][j+1] == '.':
            a,b = M*i+j,M*i+(j+1)
            if (i+j) % 2 == 1:
                a,b = b,a
            D.add_edge(a,b,1)

print(D.flow(s,t))

ANS = deepcopy(S)
for a in range(N*M):
    for b,cap,_ in D.G[a]:
        if b == s or b == t:
            continue
        ai,aj = divmod(a,M)
        bi,bj = divmod(b,M)
        if (ai+aj) % 2 == 0 and cap == 0:
            if ai+1 == bi:
                ANS[ai][aj] = 'v'
                ANS[bi][bj] = '^'
            elif ai == bi+1:
                ANS[ai][aj] = '^'
                ANS[bi][bj] = 'v'
            elif aj+1 == bj:
                ANS[ai][aj] = '>'
                ANS[bi][bj] = '<'
            elif aj == bj+1:
                ANS[ai][aj] = '<'
                ANS[bi][bj] = '>'

for i in range(N):
    print(''.join(ANS[i]))

Solution (Python):
import sys
from collections import deque

sys.setrecursionlimit(10 ** 7)
input = sys.stdin.readline
f_inf = float('inf')
mod = 10 ** 9 + 7


class Dinic:
    def __init__(self, N):
        self.N = N
        self.G = [[] for _ in range(N)]

    def add_edge(self, fr, to, cap):
        forward = [to, cap, None]
        forward[2] = backward = [fr, 0, forward]
        self.G[fr].append(forward)
        self.G[to].append(backward)

    def add_multi_edge(self, v1, v2, cap1, cap2):
        edge1 = [v2, cap1, None]
        edge1[2] = edge2 = [v1, cap2, edge1]
        self.G[v1].append(edge1)
        self.G[v2].append(edge2)

    def bfs(self, s, t):
        self.level = [None] * self.N
        self.level[s] = 0
        que = deque([s])
        G = self.G
        while que:
            v = que.popleft()
            next_level = self.level[v] + 1
            for w, cap, _ in G[v]:
                if cap and self.level[w] is None:
                    self.level[w] = next_level
                    que.append(w)
        return self.level[t] is not None

    def dfs(self, v, t, f):
        if v == t:
            return f
        level = self.level
        for e in self.it[v]:
            w, cap, rev = e
            if cap and level[v] < level[w]:
                d = self.dfs(w, t, min(f, cap))
                if d:
                    e[1] -= d
                    rev[1] += d
                    return d
        return 0

    def flow(self, s, t):
        flow = 0
        INF = 10 ** 9 + 7
        G = self.G
        while self.bfs(s, t):
            *self.it, = map(iter, self.G)
            f = INF
            while f:
                f = self.dfs(s, t, INF)
                flow += f
        return flow


def resolve():
    n, m = map(int, input().split())
    grid = [list(input().rstrip()) for _ in range(n)]

    mf = Dinic(n * m + 2)
    start = n * m
    end = start + 1

    for i in range(n):
        for j in range(m):
            if grid[i][j] == "
                continue
            if (i + j) % 2 == 0:
                mf.add_edge(start, i * m + j, 1)
                for di, dj in [(0, 1), (1, 0), (-1, 0), (0, -1)]:
                    next_i, next_j = i + di, j + dj
                    if next_i < 0 or next_i >= n or next_j < 0 or next_j >= m:
                        continue
                    if grid[next_i][next_j] == "
                        continue
                    mf.add_edge(i * m + j, next_i * m + next_j, 1)
            else:
                mf.add_edge(i * m + j, end, 1)

    res = mf.flow(start, end)

    for y in range(n):
        for x in range(m):
            if grid[y][x] == "
                continue
            for to, cap, _ in mf.G[y * m + x]:
                if cap == 0 and to != start and to != end:
                    next_y, next_x = divmod(to, m)
                    if y + 1 == next_y:
                        grid[y][x] = 'v'
                        grid[next_y][next_x] = '^'
                    elif y == next_y + 1:
                        grid[y][x] = '^'
                        grid[next_y][next_x] = 'v'
                    elif x + 1 == next_x:
                        grid[y][x] = '>'
                        grid[next_y][next_x] = '<'
                    elif x == next_x + 1:
                        grid[y][x] = '<'
                        grid[next_y][next_x] = '>'

    print(res)
    print(*[''.join(ret) for ret in grid], sep="\n")


if __name__ == '__main__':
    resolve()

Solution (Python):
import sys
from collections import deque

sys.setrecursionlimit(10 ** 7)
input = sys.stdin.readline
f_inf = float('inf')
mod = 10 ** 9 + 7


class Dinic:
    def __init__(self, N):
        self.N = N
        self.G = [[] for _ in range(N)]

    def add_edge(self, fr, to, cap):
        forward = [to, cap, None]
        forward[2] = backward = [fr, 0, forward]
        self.G[fr].append(forward)
        self.G[to].append(backward)

    def add_multi_edge(self, v1, v2, cap1, cap2):
        edge1 = [v2, cap1, None]
        edge1[2] = edge2 = [v1, cap2, edge1]
        self.G[v1].append(edge1)
        self.G[v2].append(edge2)

    def bfs(self, s, t):
        self.level = [None] * self.N
        self.level[s] = 0
        que = deque([s])
        G = self.G
        while que:
            v = que.popleft()
            next_level = self.level[v] + 1
            for w, cap, _ in G[v]:
                if cap and self.level[w] is None:
                    self.level[w] = next_level
                    que.append(w)
        return self.level[t] is not None

    def dfs(self, v, t, f):
        if v == t:
            return f
        level = self.level
        for e in self.it[v]:
            w, cap, rev = e
            if cap and level[v] < level[w]:
                d = self.dfs(w, t, min(f, cap))
                if d:
                    e[1] -= d
                    rev[1] += d
                    return d
        return 0

    def flow(self, s, t):
        flow = 0
        INF = 10 ** 9 + 7
        G = self.G
        while self.bfs(s, t):
            *self.it, = map(iter, self.G)
            f = INF
            while f:
                f = self.dfs(s, t, INF)
                flow += f
        return flow


def resolve():
    n, m = map(int, input().split())
    grid = [list(input().rstrip()) for _ in range(n)]

    mf = Dinic(n * m + 2)
    start = n * m
    end = start + 1
    
    for i in range(n):
        for j in range(m):
            if grid[i][j] == "
                continue
            if (i + j) % 2 == 0:
                mf.add_edge(start, i * m + j, 1)
                for di, dj in [(0, 1), (1, 0), (-1, 0), (0, -1)]:
                    next_i, next_j = i + di, j + dj
                    if next_i < 0 or next_i >= n or next_j < 0 or next_j >= m:
                        continue
                    if grid[next_i][next_j] == "
                        continue
                    mf.add_edge(i * m + j, next_i * m + next_j, 1)
            else:
                mf.add_edge(i * m + j, end, 1)

    res = mf.flow(start, end)

    for y in range(n):
        for x in range(m):
            for to, cap, rev in mf.G[y * m + x]:
                next_y, next_x = divmod(to, m)
                if (y + x) % 2 == 0 and cap == 0 and to != start and to != end and \
                        (y * m + x) != start and (y * m + x) != end:
                    if y + 1 == next_y:
                        grid[y][x] = 'v'
                        grid[next_y][next_x] = '^'
                    elif y == next_y + 1:
                        grid[y][x] = '^'
                        grid[next_y][next_x] = 'v'
                    elif x + 1 == next_x:
                        grid[y][x] = '>'
                        grid[next_y][next_x] = '<'
                    elif x == next_x + 1:
                        grid[y][x] = '<'
                        grid[next_y][next_x] = '>'

    print(res)
    print(*[''.join(ret) for ret in grid], sep="\n")


if __name__ == '__main__':
    resolve()