Language: C++

#include <bits/stdc++.h>
using namespace std;

template <class Cap> struct mf_graph {
  public:
    mf_graph() : _n(0) {}
    mf_graph(int n) : _n(n), g(n) {}

    int add_edge(int from, int to, Cap cap) {
        assert(0 <= from && from < _n);
        assert(0 <= to && to < _n);
        assert(0 <= cap);
        int m = int(pos.size());
        pos.push_back({from, int(g[from].size())});
        g[from].push_back(_edge{to, int(g[to].size()), cap});
        g[to].push_back(_edge{from, int(g[from].size()) - 1, 0});
        return m;
    }

    struct edge {
        int from, to;
        Cap cap, flow;
    };

    edge get_edge(int i) {
        int m = int(pos.size());
        assert(0 <= i && i < m);
        auto _e = g[pos[i].first][pos[i].second];
        auto _re = g[_e.to][_e.rev];
        return edge{pos[i].first, _e.to, _e.cap + _re.cap, _re.cap};
    }
    std::vector<edge> edges() {
        int m = int(pos.size());
        std::vector<edge> result;
        for (int i = 0; i < m; i++) {
            result.push_back(get_edge(i));
        }
        return result;
    }
    void change_edge(int i, Cap new_cap, Cap new_flow) {
        int m = int(pos.size());
        assert(0 <= i && i < m);
        assert(0 <= new_flow && new_flow <= new_cap);
        auto& _e = g[pos[i].first][pos[i].second];
        auto& _re = g[_e.to][_e.rev];
        _e.cap = new_cap - new_flow;
        _re.cap = new_flow;
    }

    Cap flow(int s, int t) {
        return flow(s, t, std::numeric_limits<Cap>::max());
    }
    Cap flow(int s, int t, Cap flow_limit) {
        assert(0 <= s && s < _n);
        assert(0 <= t && t < _n);

        std::vector<int> level(_n), iter(_n);
        internal::simple_queue<int> que;

        auto bfs = [&]() {
            std::fill(level.begin(), level.end(), -1);
            level[s] = 0;
            que.clear();
            que.push(s);
            while (!que.empty()) {
                int v = que.front();
                que.pop();
                for (auto e : g[v]) {
                    if (e.cap == 0 || level[e.to] >= 0) continue;
                    level[e.to] = level[v] + 1;
                    if (e.to == t) return;
                    que.push(e.to);
                }
            }
        };
        auto dfs = [&](auto self, int v, Cap up) {
            if (v == s) return up;
            Cap res = 0;
            int level_v = level[v];
            for (int& i = iter[v]; i < int(g[v].size()); i++) {
                _edge& e = g[v][i];
                if (level_v <= level[e.to] || g[e.to][e.rev].cap == 0) continue;
                Cap d =
                    self(self, e.to, std::min(up - res, g[e.to][e.rev].cap));
                if (d <= 0) continue;
                g[v][i].cap += d;
                g[e.to][e.rev].cap -= d;
                res += d;
                if (res == up) break;
            }
            return res;
        };

        Cap flow = 0;
        while (flow < flow_limit) {
            bfs();
            if (level[t] == -1) break;
            std::fill(iter.begin(), iter.end(), 0);
            while (flow < flow_limit) {
                Cap f = dfs(dfs, t, flow_limit - flow);
                if (!f) break;
                flow += f;
            }
        }
        return flow;
    }

    std::vector<bool> min_cut(int s) {
        std::vector<bool> visited(_n);
        internal::simple_queue<int> que;
        que.push(s);
        while (!que.empty()) {
            int p = que.front();
            que.pop();
            visited[p] = true;
            for (auto e : g[p]) {
                if (e.cap && !visited[e.to]) {
                    visited[e.to] = true;
                    que.push(e.to);
                }
            }
        }
        return visited;
    }

  private:
    int _n;
    struct _edge {
        int to, rev;
        Cap cap;
    };
    std::vector<std::pair<int, int>> pos;
    std::vector<std::vector<_edge>> g;
};

const int di[4] = {1,0,-1,0};
const int dj[4] = {0,1,0,-1};
 
int main(){
    int n, m; cin >> n >> m;
    vector<string> data(n);
    for (int i = 0; i < n; i++) cin >> data[i];
 
    mf_graph<int> ans(n * m + 2);
    int s = n * m, t = s + 1;
    for (int i = 0; i < n; i++) {
        for (int j = 0; j < m; j++) {
            if (data[i][j] == '#') continue;
            if ((i + j) % 2 == 1) {
                ans.add_edge(m * i + j, t, 1);
            }
            else {
                ans.add_edge(s, m * i + j, 1);
                for (int k = 0; k < 4; k++) {
                    int ni = i + di[k], nj = j + dj[k];
                    if (0 <= ni && ni < n && 0 <= nj && nj < m && data[ni][nj] == '.') {
                        ans.add_edge(m * i + j, m * ni + nj, 1);
                    }
                }
            }
        }
    }
 
    cout << ans.flow(s, t) << "\n";
 
    for (auto p : ans.edges()) {
        if (p.from == s || p.to == t || p.flow == 0) continue;
        int x = p.from / m;
        int y = p.from % m;
        int nx = p.to / m;
        int ny = p.to % m;
        if (x + 1 == nx) {
            data[x][y] = 'v';
            data[nx][ny] = '^';
        }
        else if (y + 1 == ny) {
            data[x][y] = '>';
            data[nx][ny] = '<';
        }
        else if (x - 1 == nx) {
            data[x][y] = '^';
            data[nx][ny] = 'v';
        }
        else {
            data[x][y] = '<';
            data[nx][ny] = '>';
        }
    }
 
    for (int i = 0; i < n; i++) {
        cout << data[i] << "\n";
    }
    return 0;
}

Compiler output:
a.cc: In member function 'Cap mf_graph<Cap>::flow(int, int, Cap)':
a.cc:58:9: error: 'internal' is not a class, namespace, or enumeration
   58 |         internal::simple_queue<int> que;
      |         ^~~~~~~~
a.cc:58:32: error: expected primary-expression before 'int'
   58 |         internal::simple_queue<int> que;
      |                                ^~~
a.cc: In lambda function:
a.cc:63:13: error: 'que' was not declared in this scope
   63 |             que.clear();
      |             ^~~
a.cc: In member function 'std::vector<bool> mf_graph<Cap>::min_cut(int)':
a.cc:110:9: error: 'internal' is not a class, namespace, or enumeration
  110 |         internal::simple_queue<int> que;
      |         ^~~~~~~~
a.cc:110:32: error: expected primary-expression before 'int'
  110 |         internal::simple_queue<int> que;
      |                                ^~~
a.cc:111:9: error: 'que' was not declared in this scope
  111 |         que.push(s);
      |         ^~~