verify/graph/flow/min_cost_flow.test.cpp

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• Last update: 2026-10-06 01:57:24+09:00
• Problem: https://onlinejudge.u-aizu.ac.jp/problems/GRL_6_B

Depends on

• Bounded Min Cost Flow (graph/flow/bounded_min_cost_flow.hpp)
• Max Flow (graph/flow/max_flow.hpp)
• Min Cost Flow (graph/flow/min_cost_flow.hpp)
• Fast IO (utilities/fast_io.hpp)

Code

#define PROBLEM "https://onlinejudge.u-aizu.ac.jp/problems/GRL_6_B"

#include "../../../graph/flow/min_cost_flow.hpp"
#include "../../../utilities/fast_io.hpp"

int main() {
    m1une::utilities::FastInput input;
    m1une::utilities::FastOutput output;

    int vertex_count, edge_count;
    long long required_flow;
    input >> vertex_count >> edge_count >> required_flow;
    m1une::flow::MinCostFlow<long long, long long> flow(vertex_count);
    flow.reserve_edges(edge_count);
    for (int edge = 0; edge < edge_count; edge++) {
        int from, to;
        long long capacity, cost;
        input >> from >> to >> capacity >> cost;
        flow.add_edge(from, to, capacity, cost);
    }
    auto [sent, cost] = flow.flow(0, vertex_count - 1, required_flow);
    output << (sent == required_flow ? cost : -1) << '\n';
}

#line 1 "verify/graph/flow/min_cost_flow.test.cpp"
#define PROBLEM "https://onlinejudge.u-aizu.ac.jp/problems/GRL_6_B"

#line 1 "graph/flow/min_cost_flow.hpp"



#include <algorithm>
#include <array>
#include <bit>
#include <cassert>
#include <cstddef>
#include <functional>
#include <limits>
#include <optional>
#include <type_traits>
#include <utility>
#include <vector>

#line 1 "graph/flow/bounded_min_cost_flow.hpp"



#line 7 "graph/flow/bounded_min_cost_flow.hpp"
#include <cmath>
#line 11 "graph/flow/bounded_min_cost_flow.hpp"
#include <queue>
#line 14 "graph/flow/bounded_min_cost_flow.hpp"

namespace m1une {
namespace flow {

template <
    class Cap,
    class Cost,
    class TotalCost = Cost,
    std::size_t PivotLimitFactor = 8
>
struct BoundedMinCostFlow {
    static_assert(std::numeric_limits<Cap>::is_integer);
    static_assert(std::numeric_limits<Cap>::is_signed);
    static_assert(std::numeric_limits<Cost>::is_specialized);
    static_assert(std::numeric_limits<Cost>::is_signed);

    struct Edge {
        int from;
        int to;
        Cap lower;
        Cap upper;
        Cost cost;
    };

    struct ResultEdge {
        int from;
        int to;
        Cap lower;
        Cap upper;
        Cap flow;
        Cost cost;
    };

    struct Result {
        std::vector<ResultEdge> edges;
        std::vector<Cap> balance;
        std::vector<Cost> potential;
        TotalCost cost;

        ResultEdge get_edge(int i) const {
            assert(0 <= i && i < int(edges.size()));
            return edges[i];
        }

        Cap flow(int i) const {
            assert(0 <= i && i < int(edges.size()));
            return edges[i].flow;
        }
    };

   private:
    struct NetworkEdge {
        int to;
        Cap cap;
        Cost cost;
    };

    struct NetworkSimplexSolver {
        enum class Status {
            optimal,
            infeasible,
            pivot_limit_reached,
        };

        struct Parent {
            int vertex;
            int edge;
            Cap up;
            Cap down;
        };

        int n;
        std::vector<NetworkEdge> edges;
        std::vector<Cap> excess;
        std::vector<Cost> potential;
        std::size_t pivot_count = 0;

        NetworkSimplexSolver(int vertex_count, const std::vector<Cap>& balance)
            : n(vertex_count), excess(balance) {}

        void reserve_edges(int edge_count) {
            edges.reserve(2 * (edge_count + n));
        }

        int add_edge(int from, int to, Cap lower, Cap upper, Cost cost) {
            int id = int(edges.size()) / 2;
            edges.push_back(NetworkEdge{to, upper - lower, cost});
            edges.push_back(NetworkEdge{from, Cap(0), -cost});
            excess[from] -= lower;
            excess[to] += lower;
            return id;
        }

        Status solve(std::size_t pivot_limit) {
            pivot_count = 0;
            const int original_edge_count = int(edges.size());
            potential.assign(n + 1, Cost(0));

            Cost artificial_cost = Cost(1);
            for (int edge = 0; edge < original_edge_count; edge += 2) {
                artificial_cost += edges[edge].cost < Cost(0)
                    ? -edges[edge].cost : edges[edge].cost;
            }

            std::vector<Parent> parent(n);
            edges.reserve(original_edge_count + 2 * n);
            for (int vertex = 0; vertex < n; vertex++) {
                if (excess[vertex] >= Cap(0)) {
                    edges.push_back(NetworkEdge{n, Cap(0), artificial_cost});
                    edges.push_back(NetworkEdge{vertex, excess[vertex], -artificial_cost});
                    potential[vertex] = -artificial_cost;
                } else {
                    edges.push_back(NetworkEdge{n, -excess[vertex], -artificial_cost});
                    edges.push_back(NetworkEdge{vertex, Cap(0), artificial_cost});
                    potential[vertex] = artificial_cost;
                }
                int edge = int(edges.size()) - 2;
                parent[vertex] = Parent{
                    n, edge, edges[edge].cap, edges[edge ^ 1].cap
                };
            }

            std::vector<int> depth(n + 1, 1);
            depth[n] = 0;
            std::vector<int> next(2 * (n + 1));
            std::vector<int> previous(2 * (n + 1));
            auto connect = [&](int first, int second) {
                next[first] = second;
                previous[second] = first;
            };
            for (int vertex = 0; vertex <= n; vertex++) {
                connect(2 * vertex, 2 * vertex + 1);
            }
            for (int vertex = 0; vertex < n; vertex++) {
                connect(2 * vertex + 1, next[2 * n]);
                connect(2 * n, 2 * vertex);
            }

            auto push_flow = [&](int entering_edge) {
                const int first = edges[entering_edge ^ 1].to;
                const int second = edges[entering_edge].to;
                const Cost cycle_cost =
                    edges[entering_edge].cost
                    + potential[first] - potential[second];

                Cap amount = edges[entering_edge].cap;
                bool leave_first_side = true;
                int leaving_vertex = second;

                int first_ancestor = first;
                int second_ancestor = second;
                auto move_first_up = [&] {
                    if (parent[first_ancestor].down < amount) {
                        amount = parent[first_ancestor].down;
                        leaving_vertex = first_ancestor;
                        leave_first_side = true;
                    }
                    first_ancestor = parent[first_ancestor].vertex;
                };
                auto move_second_up = [&] {
                    if (parent[second_ancestor].up <= amount) {
                        amount = parent[second_ancestor].up;
                        leaving_vertex = second_ancestor;
                        leave_first_side = false;
                    }
                    second_ancestor = parent[second_ancestor].vertex;
                };
                if (depth[first_ancestor] >= depth[second_ancestor]) {
                    int difference = depth[first_ancestor] - depth[second_ancestor];
                    for (int i = 0; i < difference; i++) move_first_up();
                } else {
                    int difference = depth[second_ancestor] - depth[first_ancestor];
                    for (int i = 0; i < difference; i++) move_second_up();
                }
                while (first_ancestor != second_ancestor) {
                    move_first_up();
                    move_second_up();
                }
                const int ancestor = first_ancestor;

                if (amount != Cap(0)) {
                    int vertex = first;
                    while (vertex != ancestor) {
                        parent[vertex].up += amount;
                        parent[vertex].down -= amount;
                        vertex = parent[vertex].vertex;
                    }
                    vertex = second;
                    while (vertex != ancestor) {
                        parent[vertex].up -= amount;
                        parent[vertex].down += amount;
                        vertex = parent[vertex].vertex;
                    }
                }

                int vertex = first;
                int new_parent = second;
                std::pair<Cap, Cap> parent_capacities{
                    edges[entering_edge].cap - amount,
                    edges[entering_edge ^ 1].cap + amount
                };
                Cost potential_difference = -cycle_cost;
                if (!leave_first_side) {
                    std::swap(vertex, new_parent);
                    std::swap(parent_capacities.first, parent_capacities.second);
                    potential_difference = -potential_difference;
                }
                int parent_edge = entering_edge ^ (leave_first_side ? 0 : 1);

                while (new_parent != leaving_vertex) {
                    int new_depth = depth[new_parent];
                    int tour_index = 2 * vertex;
                    while (tour_index != 2 * vertex + 1) {
                        if ((tour_index & 1) == 0) {
                            new_depth++;
                            potential[tour_index / 2] += potential_difference;
                            depth[tour_index / 2] = new_depth;
                        } else {
                            new_depth--;
                        }
                        tour_index = next[tour_index];
                    }

                    connect(previous[2 * vertex], next[2 * vertex + 1]);
                    connect(2 * vertex + 1, next[2 * new_parent]);
                    connect(2 * new_parent, 2 * vertex);

                    std::swap(parent[vertex].edge, parent_edge);
                    parent_edge ^= 1;
                    std::swap(parent[vertex].up, parent_capacities.first);
                    std::swap(parent[vertex].down, parent_capacities.second);
                    std::swap(parent_capacities.first, parent_capacities.second);

                    int old_parent = parent[vertex].vertex;
                    parent[vertex].vertex = new_parent;
                    new_parent = vertex;
                    vertex = old_parent;
                }
                edges[parent_edge].cap = parent_capacities.first;
                edges[parent_edge ^ 1].cap = parent_capacities.second;
            };

            bool pivot_limit_reached = false;
            auto pivot = [&](int entering_edge) {
                if (pivot_count == pivot_limit) {
                    pivot_limit_reached = true;
                    return false;
                }
                push_flow(entering_edge);
                pivot_count++;
                return true;
            };

            const int candidate_limit = std::max(
                int(0.2 * std::sqrt(double(original_edge_count))), 10
            );
            const int minor_limit = std::max(candidate_limit / 10, 3);
            std::vector<int> candidates;
            candidates.reserve(candidate_limit);

            auto minor_pivot = [&] {
                Cost best_cost = Cost(0);
                int best_edge = -1;
                int index = 0;
                while (index < int(candidates.size())) {
                    int edge = candidates[index];
                    if (edges[edge].cap == Cap(0)) {
                        candidates[index] = candidates.back();
                        candidates.pop_back();
                        continue;
                    }
                    Cost reduced_cost =
                        edges[edge].cost
                        + potential[edges[edge ^ 1].to]
                        - potential[edges[edge].to];
                    if (reduced_cost >= Cost(0)) {
                        candidates[index] = candidates.back();
                        candidates.pop_back();
                        continue;
                    }
                    if (reduced_cost < best_cost) {
                        best_cost = reduced_cost;
                        best_edge = edge;
                    }
                    index++;
                }
                if (best_edge == -1) return false;
                return pivot(best_edge);
            };

            int edge = 0;
            while (true) {
                for (int iteration = 0; iteration < minor_limit; iteration++) {
                    if (!minor_pivot()) break;
                }
                if (pivot_limit_reached) return Status::pivot_limit_reached;

                Cost best_cost = Cost(0);
                int best_edge = -1;
                candidates.clear();
                for (int scanned = 0; scanned < int(edges.size()); scanned++) {
                    if (edges[edge].cap != Cap(0)) {
                        Cost reduced_cost =
                            edges[edge].cost
                            + potential[edges[edge ^ 1].to]
                            - potential[edges[edge].to];
                        if (reduced_cost < Cost(0)) {
                            if (reduced_cost < best_cost) {
                                best_cost = reduced_cost;
                                best_edge = edge;
                            }
                            candidates.push_back(edge);
                            if (int(candidates.size()) == candidate_limit) break;
                        }
                    }
                    edge++;
                    if (edge == int(edges.size())) edge = 0;
                }
                if (candidates.empty()) break;
                if (!pivot(best_edge)) return Status::pivot_limit_reached;
            }

            for (int vertex = 0; vertex < n; vertex++) {
                edges[parent[vertex].edge].cap = parent[vertex].up;
                edges[parent[vertex].edge ^ 1].cap = parent[vertex].down;
            }

            bool feasible = true;
            for (int vertex = 0; vertex < n; vertex++) {
                int artificial_edge = original_edge_count + 2 * vertex;
                if (
                    (excess[vertex] >= Cap(0)
                        && edges[artificial_edge ^ 1].cap != Cap(0))
                    || (excess[vertex] < Cap(0)
                        && edges[artificial_edge].cap != Cap(0))
                ) {
                    feasible = false;
                    break;
                }
            }
            potential.pop_back();
            return feasible ? Status::optimal : Status::infeasible;
        }

        Cap edge_flow(int edge_id, Cap lower) const {
            return lower + edges[2 * edge_id + 1].cap;
        }
    };

    struct ScalingEdge {
        int to;
        int reverse;
        Cap cap;
        Cap flow;
        Cost cost;
    };

    struct ScalingSolver {
        int n;
        std::vector<std::vector<ScalingEdge>> graph;
        std::vector<std::pair<int, int>> positions;
        std::vector<Cap> excess;
        std::vector<Cost> potential;
        std::vector<Cost> distance;
        std::vector<int> parent_vertex;
        std::vector<int> parent_edge;
        std::vector<int> excess_vertices;
        std::vector<int> deficit_vertices;
        Cost farthest = Cost(0);

        ScalingSolver(int vertex_count, const std::vector<Cap>& balance)
            : n(vertex_count), graph(vertex_count), excess(balance),
              potential(vertex_count, Cost(0)) {}

        void reserve_edges(int edge_count) {
            positions.reserve(edge_count);
        }

        int add_edge(int from, int to, Cap lower, Cap upper, Cost cost) {
            int id = int(positions.size());
            int from_edge = int(graph[from].size());
            int to_edge = int(graph[to].size());
            if (from == to) to_edge++;
            positions.emplace_back(from, from_edge);
            graph[from].push_back(ScalingEdge{
                to, to_edge, upper, Cap(0), cost
            });
            graph[to].push_back(ScalingEdge{
                from, from_edge, -lower, Cap(0), -cost
            });
            return id;
        }

        Cap residual_capacity(int from, int edge_id) const {
            const auto& edge = graph[from][edge_id];
            return edge.cap - edge.flow;
        }

        Cost residual_cost(int from, const ScalingEdge& edge) const {
            return edge.cost + potential[from] - potential[edge.to];
        }

        void push(int from, int edge_id, Cap amount) {
            auto& edge = graph[from][edge_id];
            edge.flow += amount;
            graph[edge.to][edge.reverse].flow -= amount;
        }

        void saturate_negative(Cap delta) {
            excess_vertices.clear();
            deficit_vertices.clear();
            for (int from = 0; from < n; from++) {
                for (
                    int edge_id = 0;
                    edge_id < int(graph[from].size());
                    edge_id++
                ) {
                    const auto& edge = graph[from][edge_id];
                    Cap residual = edge.cap - edge.flow;
                    residual -= residual % delta;
                    if (
                        residual_cost(from, edge) < Cost(0)
                        || residual < Cap(0)
                    ) {
                        int to = edge.to;
                        push(from, edge_id, residual);
                        excess[from] -= residual;
                        excess[to] += residual;
                    }
                }
            }
            for (int vertex = 0; vertex < n; vertex++) {
                if (excess[vertex] > Cap(0)) {
                    excess_vertices.push_back(vertex);
                } else if (excess[vertex] < Cap(0)) {
                    deficit_vertices.push_back(vertex);
                }
            }
        }

        bool dual(Cap delta) {
            excess_vertices.erase(
                std::remove_if(
                    excess_vertices.begin(), excess_vertices.end(),
                    [&](int vertex) { return excess[vertex] < delta; }
                ),
                excess_vertices.end()
            );
            deficit_vertices.erase(
                std::remove_if(
                    deficit_vertices.begin(), deficit_vertices.end(),
                    [&](int vertex) { return excess[vertex] > -delta; }
                ),
                deficit_vertices.end()
            );

            const Cost unreachable = std::numeric_limits<Cost>::max();
            distance.assign(n, unreachable);
            parent_vertex.assign(n, -1);
            parent_edge.assign(n, -1);
            using QueueEntry = std::pair<Cost, int>;
            std::priority_queue<
                QueueEntry,
                std::vector<QueueEntry>,
                std::greater<QueueEntry>
            > queue;
            for (int vertex : excess_vertices) {
                distance[vertex] = Cost(0);
                queue.emplace(Cost(0), vertex);
            }

            farthest = Cost(0);
            int reached_deficits = 0;
            while (!queue.empty()) {
                auto [current_distance, from] = queue.top();
                queue.pop();
                if (distance[from] != current_distance) continue;
                farthest = current_distance;
                if (excess[from] <= -delta) reached_deficits++;
                if (reached_deficits >= int(deficit_vertices.size())) break;

                for (
                    int edge_id = 0;
                    edge_id < int(graph[from].size());
                    edge_id++
                ) {
                    const auto& edge = graph[from][edge_id];
                    if (edge.cap - edge.flow < delta) continue;
                    Cost next_distance =
                        current_distance + residual_cost(from, edge);
                    if (next_distance >= distance[edge.to]) continue;
                    distance[edge.to] = next_distance;
                    parent_vertex[edge.to] = from;
                    parent_edge[edge.to] = edge_id;
                    queue.emplace(next_distance, edge.to);
                }
            }

            for (int vertex = 0; vertex < n; vertex++) {
                potential[vertex] += std::min(distance[vertex], farthest);
            }
            return reached_deficits > 0;
        }

        void primal(Cap delta) {
            for (int sink : deficit_vertices) {
                if (distance[sink] > farthest) continue;
                Cap amount = -excess[sink];
                int root = sink;
                while (parent_edge[root] != -1) {
                    int from = parent_vertex[root];
                    amount = std::min(
                        amount,
                        residual_capacity(from, parent_edge[root])
                    );
                    root = from;
                }
                amount = std::min(amount, excess[root]);
                amount -= amount % delta;
                if (amount <= Cap(0)) continue;

                int vertex = sink;
                while (parent_edge[vertex] != -1) {
                    int from = parent_vertex[vertex];
                    int edge_id = parent_edge[vertex];
                    push(from, edge_id, amount);
                    if (residual_capacity(from, edge_id) == Cap(0)) {
                        parent_edge[vertex] = -1;
                    }
                    vertex = from;
                }
                excess[sink] += amount;
                excess[root] -= amount;
            }
        }

        bool solve() {
            Cap scale_bound = Cap(1);
            for (Cap value : excess) {
                scale_bound = std::max(scale_bound, value);
                scale_bound = std::max(scale_bound, -value);
            }
            for (const auto& edges : graph) {
                for (const auto& edge : edges) {
                    Cap residual = edge.cap - edge.flow;
                    scale_bound = std::max(scale_bound, residual);
                    scale_bound = std::max(scale_bound, -residual);
                }
            }

            Cap delta = Cap(1);
            while (delta <= scale_bound / Cap(2)) delta *= Cap(2);
            while (true) {
                saturate_negative(delta);
                while (dual(delta)) primal(delta);
                if (delta == Cap(1)) break;
                delta /= Cap(2);
            }
            return excess_vertices.empty() && deficit_vertices.empty();
        }

        Cap edge_flow(int edge_id, Cap) const {
            auto [from, index] = positions[edge_id];
            return graph[from][index].flow;
        }
    };

    int _n;
    std::vector<Edge> _edges;
    std::vector<Cap> _balance;

    template <class Solver>
    Result make_result(
        const std::vector<Cap>& balance,
        const Solver& solver,
        std::vector<Cost> potential
    ) const {
        Result result;
        result.balance = balance;
        result.cost = TotalCost(0);
        result.edges.reserve(_edges.size());
        for (int i = 0; i < int(_edges.size()); i++) {
            const auto& edge = _edges[i];
            Cap flow = solver.edge_flow(i, edge.lower);
            result.cost += TotalCost(flow) * TotalCost(edge.cost);
            result.edges.push_back(ResultEdge{
                edge.from,
                edge.to,
                edge.lower,
                edge.upper,
                flow,
                edge.cost
            });
        }
        result.potential = std::move(potential);
        return result;
    }

    std::vector<Cost> residual_potential(
        const std::vector<ResultEdge>& edges
    ) const {
        std::vector<Cost> potential(_n, Cost(0));
        bool updated = false;
        for (int iteration = 0; iteration < _n; iteration++) {
            updated = false;
            for (const ResultEdge& edge : edges) {
                if (
                    edge.flow < edge.upper
                    && potential[edge.to] > potential[edge.from] + edge.cost
                ) {
                    potential[edge.to] = potential[edge.from] + edge.cost;
                    updated = true;
                }
                if (
                    edge.lower < edge.flow
                    && potential[edge.from] > potential[edge.to] - edge.cost
                ) {
                    potential[edge.from] = potential[edge.to] - edge.cost;
                    updated = true;
                }
            }
            if (!updated) break;
        }
        assert(!updated);
        return potential;
    }

    std::optional<Result> polynomial_min_cost_flow_impl(
        const std::vector<Cap>& balance
    ) const {
        ScalingSolver solver(_n, balance);
        solver.reserve_edges(int(_edges.size()));
        for (const auto& edge : _edges) {
            solver.add_edge(
                edge.from,
                edge.to,
                edge.lower,
                edge.upper,
                edge.cost
            );
        }
        if (!solver.solve()) return std::nullopt;

        Result result = make_result(balance, solver, {});
        result.potential = residual_potential(result.edges);
        return result;
    }

   public:
    BoundedMinCostFlow() : BoundedMinCostFlow(0) {}

    explicit BoundedMinCostFlow(int n) : _n(n), _balance(n, Cap(0)) {
        assert(0 <= n);
    }

    int size() const {
        return _n;
    }

    int edge_count() const {
        return int(_edges.size());
    }

    void reserve_edges(int edge_count) {
        assert(0 <= edge_count);
        _edges.reserve(edge_count);
    }

    int add_edge(int from, int to, Cap lower, Cap upper, Cost cost) {
        assert(0 <= from && from < _n);
        assert(0 <= to && to < _n);
        assert(lower <= upper);
        int id = int(_edges.size());
        _edges.push_back(Edge{from, to, lower, upper, cost});
        return id;
    }

    Edge get_edge(int i) const {
        assert(0 <= i && i < int(_edges.size()));
        return _edges[i];
    }

    std::vector<Edge> edges() const {
        return _edges;
    }

    void set_balance(int v, Cap b) {
        assert(0 <= v && v < _n);
        _balance[v] = b;
    }

    void add_balance(int v, Cap b) {
        assert(0 <= v && v < _n);
        _balance[v] += b;
    }

    void add_supply(int v, Cap supply) {
        assert(Cap(0) <= supply);
        add_balance(v, supply);
    }

    void add_demand(int v, Cap demand) {
        assert(Cap(0) <= demand);
        add_balance(v, -demand);
    }

    Cap balance(int v) const {
        assert(0 <= v && v < _n);
        return _balance[v];
    }

    const std::vector<Cap>& balances() const {
        return _balance;
    }

    std::optional<Result> min_cost_flow() const {
        return min_cost_flow(_balance);
    }

    std::optional<Result> min_cost_flow(const std::vector<Cap>& balance) const {
        assert(int(balance.size()) == _n);
        Cap balance_sum = Cap(0);
        for (Cap value : balance) balance_sum += value;
        if (balance_sum != Cap(0)) return std::nullopt;

        NetworkSimplexSolver solver(_n, balance);
        solver.reserve_edges(int(_edges.size()));
        for (const auto& edge : _edges) {
            solver.add_edge(edge.from, edge.to, edge.lower, edge.upper, edge.cost);
        }
        const std::size_t graph_size =
            std::size_t(_n) + _edges.size() + 1;
        std::size_t pivot_limit = 0;
        if constexpr (PivotLimitFactor != 0) {
            const std::size_t maximum =
                std::numeric_limits<std::size_t>::max();
            pivot_limit = graph_size > maximum / PivotLimitFactor
                ? maximum : PivotLimitFactor * graph_size;
        }
        auto status = solver.solve(pivot_limit);
        if (status == NetworkSimplexSolver::Status::infeasible) {
            return std::nullopt;
        }
        if (status == NetworkSimplexSolver::Status::pivot_limit_reached) {
            return polynomial_min_cost_flow_impl(balance);
        }
        return make_result(balance, solver, std::move(solver.potential));
    }

    std::optional<Result> min_cost_flow_polynomial() const {
        return min_cost_flow_polynomial(_balance);
    }

    std::optional<Result> min_cost_flow_polynomial(
        const std::vector<Cap>& balance
    ) const {
        assert(int(balance.size()) == _n);
        Cap balance_sum = Cap(0);
        for (Cap value : balance) balance_sum += value;
        if (balance_sum != Cap(0)) return std::nullopt;
        return polynomial_min_cost_flow_impl(balance);
    }

    std::optional<Result> min_cost_st_flow(int s, int t, Cap flow_value) const {
        assert(0 <= s && s < _n);
        assert(0 <= t && t < _n);
        assert(s != t);
        std::vector<Cap> balance = _balance;
        balance[s] += flow_value;
        balance[t] -= flow_value;
        return min_cost_flow(balance);
    }

    std::optional<Result> min_cost_st_flow_polynomial(
        int s,
        int t,
        Cap flow_value
    ) const {
        assert(0 <= s && s < _n);
        assert(0 <= t && t < _n);
        assert(s != t);
        std::vector<Cap> balance = _balance;
        balance[s] += flow_value;
        balance[t] -= flow_value;
        return min_cost_flow_polynomial(balance);
    }
};

template <
    class Cap,
    class Cost,
    class TotalCost = Cost,
    std::size_t PivotLimitFactor = 8
>
using BMinCostFlow = BoundedMinCostFlow<
    Cap,
    Cost,
    TotalCost,
    PivotLimitFactor
>;

}  // namespace flow
}  // namespace m1une


#line 1 "graph/flow/max_flow.hpp"



#line 9 "graph/flow/max_flow.hpp"

namespace m1une {
namespace flow {

template <class Cap>
struct MaxFlow {
    struct Edge {
        int from;
        int to;
        Cap cap;
        Cap flow;
    };

   private:
    struct InternalEdge {
        int to;
        int rev;
        Cap cap;
    };

    struct Position {
        int from;
        int edge;
    };

    int _n;
    std::vector<Position> _pos;
    std::vector<std::vector<InternalEdge>> _g;

    Cap highest_label_preflow_push(int s, int t) {
        const int dead = 2 * _n;
        const int unreachable = _n + 1;
        std::vector<Cap> excess(_n, Cap(0));
        std::vector<int> state(8 * std::size_t(_n) + 2);
        int* height = state.data();
        int* height_count = height + _n;
        int* current = height_count + dead + 1;
        int* queue = current + _n;
        int* next = queue + _n;
        int* bucket_head = next + _n;
        std::vector<char> active(_n, false);
        int highest = -1;
        long long work = 0;
        const long long arc_count =
            2LL * static_cast<long long>(_pos.size());
        const long long work_limit = std::max(1LL, 4 * arc_count + _n);

        auto activate = [&](int v) {
            if (v == s || v == t || active[v] || excess[v] == Cap(0) ||
                height[v] >= dead) {
                return;
            }
            active[v] = true;
            next[v] = bucket_head[height[v]];
            bucket_head[height[v]] = v;
            highest = std::max(highest, height[v]);
        };

        auto rebuild_buckets = [&]() {
            std::fill(bucket_head, bucket_head + dead + 1, -1);
            std::fill(active.begin(), active.end(), false);
            highest = -1;
            for (int v = 0; v < _n; v++) activate(v);
        };

        auto global_relabel = [&]() {
            std::fill(height, height + _n, unreachable);
            std::fill(height_count, height_count + dead + 1, 0);
            std::fill(current, current + _n, 0);
            int head = 0;
            int tail = 0;
            height[t] = 0;
            height[s] = _n;
            queue[tail++] = t;
            while (head != tail) {
                int v = queue[head++];
                for (const auto& e : _g[v]) {
                    if (e.to == s || height[e.to] != unreachable) continue;
                    const auto& reverse = _g[e.to][e.rev];
                    if (reverse.cap == Cap(0)) continue;
                    height[e.to] = height[v] + 1;
                    queue[tail++] = e.to;
                }
            }
            for (int v = 0; v < _n; v++) height_count[height[v]]++;
            rebuild_buckets();
            work = 0;
        };

        auto gap = [&](int empty_height) {
            for (int v = 0; v < _n; v++) {
                if (v == s || v == t || height[v] <= empty_height ||
                    height[v] >= _n) {
                    continue;
                }
                height_count[height[v]]--;
                height[v] = unreachable;
                height_count[height[v]]++;
                current[v] = 0;
            }
            rebuild_buckets();
        };

        auto relabel = [&](int v) -> bool {
            int old_height = height[v];
            int new_height = dead;
            work += int(_g[v].size());
            for (const auto& e : _g[v]) {
                if (e.cap != Cap(0)) {
                    new_height = std::min(new_height, height[e.to] + 1);
                }
            }
            height_count[old_height]--;
            height[v] = std::min(new_height, dead);
            height_count[height[v]]++;
            current[v] = 0;
            if (old_height < _n && height_count[old_height] == 0) {
                gap(old_height);
                return true;
            }
            return false;
        };

        auto push = [&](int v, InternalEdge& e) {
            Cap sent = std::min(excess[v], e.cap);
            bool was_zero = excess[e.to] == Cap(0);
            e.cap -= sent;
            _g[e.to][e.rev].cap += sent;
            excess[v] -= sent;
            excess[e.to] += sent;
            if (was_zero) activate(e.to);
        };

        auto discharge = [&](int v) {
            while (excess[v] != Cap(0) && height[v] < dead) {
                if (current[v] == int(_g[v].size())) {
                    if (relabel(v)) return;
                    continue;
                }
                auto& e = _g[v][current[v]];
                work++;
                if (e.cap != Cap(0) && height[v] == height[e.to] + 1) {
                    push(v, e);
                } else {
                    current[v]++;
                }
            }
            activate(v);
        };

        for (auto& e : _g[s]) {
            if (e.to == s || e.cap == Cap(0)) continue;
            Cap sent = e.cap;
            e.cap = Cap(0);
            _g[e.to][e.rev].cap += sent;
            excess[e.to] += sent;
        }
        global_relabel();

        while (highest >= 0) {
            if (bucket_head[highest] == -1) {
                highest--;
                continue;
            }
            int v = bucket_head[highest];
            bucket_head[highest] = next[v];
            if (!active[v] || height[v] != highest) continue;
            active[v] = false;
            discharge(v);
            if (work >= work_limit) global_relabel();
        }
        return excess[t];
    }

   public:
    MaxFlow() : MaxFlow(0) {}

    explicit MaxFlow(int n) : _n(n), _g(n) {
        assert(0 <= n);
    }

    int size() const {
        return _n;
    }

    int edge_count() const {
        return int(_pos.size());
    }

    void reserve_edges(int edge_count) {
        assert(0 <= edge_count);
        _pos.reserve(edge_count);
        if (_n == 0 || edge_count == 0 ||
            2 * std::size_t(edge_count) < std::size_t(_n)) {
            return;
        }
        const std::size_t average_degree =
            (3 * std::size_t(edge_count) + std::size_t(_n) - 1)
            / std::size_t(_n);
        for (auto& edges : _g) edges.reserve(average_degree);
    }

    void reserve_edges(int edge_count, const std::vector<int>& degrees) {
        assert(0 <= edge_count);
        assert(int(degrees.size()) == _n);
        _pos.reserve(edge_count);
        for (int v = 0; v < _n; v++) {
            assert(0 <= degrees[v]);
            _g[v].reserve(degrees[v]);
        }
    }

    int add_edge(int from, int to, Cap cap) {
        assert(0 <= from && from < _n);
        assert(0 <= to && to < _n);
        assert(Cap(0) <= cap);
        int id = int(_pos.size());
        int from_id = int(_g[from].size());
        int to_id = int(_g[to].size());
        if (from == to) to_id++;
        _pos.push_back(Position{from, from_id});
        _g[from].push_back(InternalEdge{to, to_id, cap});
        _g[to].push_back(InternalEdge{from, from_id, Cap(0)});
        return id;
    }

    int add_undirected_edge(int first, int second, Cap cap) {
        static_assert(std::numeric_limits<Cap>::is_signed);
        assert(0 <= first && first < _n);
        assert(0 <= second && second < _n);
        assert(Cap(0) <= cap);
        assert(cap <= std::numeric_limits<Cap>::max() / Cap(2));
        int id = int(_pos.size());
        int first_id = int(_g[first].size());
        int second_id = int(_g[second].size());
        if (first == second) second_id++;
        _pos.push_back(Position{first, ~first_id});
        _g[first].push_back(InternalEdge{second, second_id, cap});
        _g[second].push_back(InternalEdge{first, first_id, cap});
        return id;
    }

    Edge get_edge(int i) const {
        assert(0 <= i && i < int(_pos.size()));
        const auto& position = _pos[i];
        int from = position.from;
        bool undirected = position.edge < 0;
        int idx = undirected ? ~position.edge : position.edge;
        const auto& e = _g[from][idx];
        const auto& re = _g[e.to][e.rev];
        if (undirected) {
            return Edge{
                from,
                e.to,
                (e.cap + re.cap) / Cap(2),
                (re.cap - e.cap) / Cap(2)
            };
        }
        return Edge{from, e.to, e.cap + re.cap, re.cap};
    }

    std::vector<Edge> edges() const {
        std::vector<Edge> result;
        result.reserve(_pos.size());
        for (int i = 0; i < int(_pos.size()); i++) result.push_back(get_edge(i));
        return result;
    }

    void change_edge(int i, Cap new_cap, Cap new_flow) {
        assert(0 <= i && i < int(_pos.size()));
        assert(Cap(0) <= new_cap);
        auto& position = _pos[i];
        int from = position.from;
        bool undirected = position.edge < 0;
        int idx = undirected ? ~position.edge : position.edge;
        auto& e = _g[from][idx];
        auto& re = _g[e.to][e.rev];
        if (undirected) {
            assert(new_cap <= std::numeric_limits<Cap>::max() / Cap(2));
            assert(-new_cap <= new_flow && new_flow <= new_cap);
            e.cap = new_cap - new_flow;
            re.cap = new_cap + new_flow;
        } else {
            assert(Cap(0) <= new_flow && new_flow <= new_cap);
            e.cap = new_cap - new_flow;
            re.cap = new_flow;
        }
    }

    Cap max_flow(int s, int t) {
        assert(0 <= s && s < _n);
        assert(0 <= t && t < _n);
        assert(s != t);
        return highest_label_preflow_push(s, t);
    }

    Cap max_flow_push_relabel(int s, int t) {
        assert(0 <= s && s < _n);
        assert(0 <= t && t < _n);
        assert(s != t);
        return highest_label_preflow_push(s, t);
    }

    Cap max_flow_dinic(int s, int t) {
        return max_flow(s, t, std::numeric_limits<Cap>::max());
    }

    Cap max_flow(int s, int t, Cap flow_limit) {
        assert(0 <= s && s < _n);
        assert(0 <= t && t < _n);
        assert(s != t);

        std::vector<int> work(3 * std::size_t(_n));
        int* level = work.data();
        int* iter = level + _n;
        int* queue = iter + _n;
        auto bfs = [&]() -> bool {
            std::fill(level, level + _n, -1);
            int head = 0;
            int tail = 0;
            level[s] = 0;
            queue[tail++] = s;
            while (head != tail) {
                int v = queue[head++];
                for (const auto& e : _g[v]) {
                    if (level[e.to] != -1 || e.cap == Cap(0)) continue;
                    level[e.to] = level[v] + 1;
                    if (e.to == t) return true;
                    queue[tail++] = e.to;
                }
            }
            return level[t] != -1;
        };

        auto dfs = [&](auto&& self, int v, Cap up) -> Cap {
            if (v == s) return up;
            Cap result = Cap(0);
            const int current_level = level[v];
            auto& edges = _g[v];
            const int edge_count = int(edges.size());
            for (int& i = iter[v]; i < edge_count; i++) {
                auto& e = edges[i];
                if (level[e.to] + 1 != current_level) continue;
                auto& reverse = _g[e.to][e.rev];
                if (reverse.cap == Cap(0)) continue;
                Cap d = self(
                    self,
                    e.to,
                    std::min(up - result, reverse.cap)
                );
                if (d == Cap(0)) continue;
                e.cap += d;
                reverse.cap -= d;
                result += d;
                if (result == up) return result;
            }
            level[v] = _n;
            return result;
        };

        Cap flow = 0;
        while (flow < flow_limit && bfs()) {
            std::fill(iter, iter + _n, 0);
            flow += dfs(dfs, t, flow_limit - flow);
        }
        return flow;
    }

    std::vector<bool> min_cut(int s) const {
        assert(0 <= s && s < _n);
        std::vector<bool> visited(_n, false);
        std::vector<int> queue(_n);
        int head = 0;
        int tail = 0;
        visited[s] = true;
        queue[tail++] = s;
        while (head != tail) {
            int v = queue[head++];
            for (const auto& e : _g[v]) {
                if (e.cap == Cap(0) || visited[e.to]) continue;
                visited[e.to] = true;
                queue[tail++] = e.to;
            }
        }
        return visited;
    }
};

}  // namespace flow
}  // namespace m1une


#line 18 "graph/flow/min_cost_flow.hpp"

namespace m1une {
namespace flow {

template <class Cap, class Cost>
struct MinCostFlow {
    struct Edge {
        int from;
        int to;
        Cap cap;
        Cap flow;
        Cost cost;
    };

   private:
    struct InternalEdge {
        int to;
        int rev;
        Cap cap;
        Cost cost;
    };

    int _n;
    std::vector<std::pair<int, int>> _pos;
    std::vector<std::vector<InternalEdge>> _g;
    bool _has_negative_cost;
    bool _has_flow;

    template <class Key>
    struct RadixHeap {
        using Unsigned = std::make_unsigned_t<Key>;
        static constexpr int bits = std::numeric_limits<Unsigned>::digits;

        std::array<std::vector<std::pair<Unsigned, int>>, bits + 1> bucket;
        Unsigned last = 0;
        std::size_t count = 0;

        static int index(Unsigned first, Unsigned second) {
            return int(std::bit_width(first ^ second));
        }

        void clear() {
            for (auto& values : bucket) values.clear();
            last = 0;
            count = 0;
        }

        bool empty() const {
            return count == 0;
        }

        void push(Key key, int vertex) {
            Unsigned value = static_cast<Unsigned>(key);
            assert(last <= value);
            bucket[index(value, last)].emplace_back(value, vertex);
            count++;
        }

        std::pair<Key, int> pop() {
            if (bucket[0].empty()) {
                int i = 1;
                while (bucket[i].empty()) i++;
                last = bucket[i][0].first;
                for (const auto& value : bucket[i]) {
                    last = std::min(last, value.first);
                }
                for (const auto& value : bucket[i]) {
                    bucket[index(value.first, last)].push_back(value);
                }
                bucket[i].clear();
            }
            auto [key, vertex] = bucket[0].back();
            bucket[0].pop_back();
            count--;
            return {static_cast<Key>(key), vertex};
        }
    };

    template <class Key>
    struct BinaryHeap {
        using Value = std::pair<Key, int>;
        std::vector<Value> heap;

        void clear() {
            heap.clear();
        }

        bool empty() const {
            return heap.empty();
        }

        void push(Key key, int vertex) {
            heap.emplace_back(key, vertex);
            std::push_heap(heap.begin(), heap.end(), std::greater<Value>());
        }

        Value pop() {
            std::pop_heap(heap.begin(), heap.end(), std::greater<Value>());
            Value result = heap.back();
            heap.pop_back();
            return result;
        }
    };

    template <
        class Key,
        bool UseRadix =
            std::numeric_limits<Key>::is_integer && sizeof(Key) <= 8
    >
    struct HeapSelector {
        using Type = BinaryHeap<Key>;
    };

    template <class Key>
    struct HeapSelector<Key, true> {
        using Type = RadixHeap<Key>;
    };

    bool use_network_simplex(int s, int t, Cap flow_limit) const {
        if (_has_negative_cost) return false;
        if (_pos.size() < 64) return false;
        auto add_saturated = [](Cap first, Cap second) {
            const Cap maximum = std::numeric_limits<Cap>::max();
            return maximum - first < second ? maximum : first + second;
        };
        struct TerminalCapacity {
            Cap total = Cap(0);
            std::array<Cap, 7> largest{};
        };
        auto add_capacity = [&](TerminalCapacity& terminal, Cap cap) {
            terminal.total = add_saturated(terminal.total, cap);
            for (Cap& current : terminal.largest) {
                if (cap <= current) break;
                std::swap(cap, current);
            }
        };
        TerminalCapacity source;
        for (const auto& e : _g[s]) {
            if (e.to == s) continue;
            add_capacity(source, e.cap);
        }
        TerminalCapacity sink;
        for (const auto& e : _g[t]) {
            if (e.to == t) continue;
            Cap cap = _g[e.to][e.rev].cap;
            add_capacity(sink, cap);
        }
        Cap target = std::min(
            flow_limit,
            std::min(source.total, sink.total)
        );
        if (target == Cap(0)) return false;
        auto requires_eight_arcs = [&](const TerminalCapacity& terminal) {
            Cap sum = Cap(0);
            for (Cap cap : terminal.largest) {
                sum = add_saturated(sum, cap);
            }
            return sum < target;
        };
        return requires_eight_arcs(source) && requires_eight_arcs(sink);
    }

    std::pair<Cap, Cost> network_simplex_flow(
        int s,
        int t,
        Cap flow_limit
    ) {
        struct ResidualArc {
            int edge;
            bool reverse;
        };

        using Solver = BoundedMinCostFlow<Cap, Cost, Cost>;
        std::vector<ResidualArc> arcs;
        arcs.reserve(2 * _pos.size());
        for (int i = 0; i < int(_pos.size()); i++) {
            auto [from, idx] = _pos[i];
            const auto& e = _g[from][idx];
            const auto& reverse = _g[e.to][e.rev];
            if (e.cap != Cap(0)) {
                arcs.push_back(ResidualArc{i, false});
            }
            if (reverse.cap != Cap(0)) {
                arcs.push_back(ResidualArc{i, true});
            }
        }

        auto add_saturated = [](Cap first, Cap second, bool& exact) {
            const Cap maximum = std::numeric_limits<Cap>::max();
            if (maximum - first < second) {
                exact = false;
                return maximum;
            }
            return first + second;
        };
        bool source_capacity_exact = true;
        Cap source_capacity = Cap(0);
        for (const auto& e : _g[s]) {
            if (e.to == s) continue;
            source_capacity = add_saturated(
                source_capacity,
                e.cap,
                source_capacity_exact
            );
        }
        bool sink_capacity_exact = true;
        Cap sink_capacity = Cap(0);
        for (const auto& e : _g[t]) {
            if (e.to == t) continue;
            sink_capacity = add_saturated(
                sink_capacity,
                _g[e.to][e.rev].cap,
                sink_capacity_exact
            );
        }
        Cap target = std::min(
            flow_limit,
            std::min(source_capacity, sink_capacity)
        );
        if (target == Cap(0)) return {Cap(0), Cost(0)};

        struct ArcData {
            int from;
            int to;
            Cap cap;
            Cost cost;
        };
        auto arc_data = [&](const ResidualArc& arc) {
            auto [from, idx] = _pos[arc.edge];
            const auto& e = _g[from][idx];
            const auto& reverse = _g[e.to][e.rev];
            return arc.reverse
                ? ArcData{e.to, from, reverse.cap, reverse.cost}
                : ArcData{from, e.to, e.cap, e.cost};
        };
        auto apply_flow = [&](const ResidualArc& arc, Cap amount) {
            auto [from, idx] = _pos[arc.edge];
            auto& e = _g[from][idx];
            auto& reverse = _g[e.to][e.rev];
            if (arc.reverse) {
                reverse.cap -= amount;
                e.cap += amount;
            } else {
                e.cap -= amount;
                reverse.cap += amount;
            }
        };

        bool target_infeasible = false;
        if (
            source_capacity_exact && sink_capacity_exact &&
            target == source_capacity && target == sink_capacity
        ) {
            Solver terminal_solver(_n);
            terminal_solver.reserve_edges(int(arcs.size()));
            std::vector<Cap> balance(_n, Cap(0));
            std::vector<int> internal_arcs;
            std::vector<int> fixed_arcs;
            internal_arcs.reserve(arcs.size());
            fixed_arcs.reserve(_g[s].size() + _g[t].size());
            Cost fixed_cost = Cost(0);
            for (int i = 0; i < int(arcs.size()); i++) {
                ArcData data = arc_data(arcs[i]);
                if (data.from == s) {
                    if (data.to == s) continue;
                    fixed_arcs.push_back(i);
                    fixed_cost += Cost(data.cap) * data.cost;
                    if (data.to != t) balance[data.to] += data.cap;
                } else if (data.to == t) {
                    if (data.from == t) continue;
                    fixed_arcs.push_back(i);
                    fixed_cost += Cost(data.cap) * data.cost;
                    balance[data.from] -= data.cap;
                } else if (data.to != s && data.from != t) {
                    terminal_solver.add_edge(
                        data.from,
                        data.to,
                        Cap(0),
                        data.cap,
                        data.cost
                    );
                    internal_arcs.push_back(i);
                }
            }
            auto terminal_result = terminal_solver.min_cost_flow(balance);
            if (terminal_result) {
                for (int i : fixed_arcs) {
                    apply_flow(arcs[i], arc_data(arcs[i]).cap);
                }
                for (int i = 0; i < int(internal_arcs.size()); i++) {
                    apply_flow(
                        arcs[internal_arcs[i]],
                        terminal_result->flow(i)
                    );
                }
                _has_flow = true;
                return {target, fixed_cost + terminal_result->cost};
            }
            target_infeasible = true;
        }

        Solver solver(_n);
        solver.reserve_edges(int(arcs.size()));
        for (const auto& arc : arcs) {
            ArcData data = arc_data(arc);
            solver.add_edge(
                data.from,
                data.to,
                Cap(0),
                data.cap,
                data.cost
            );
        }
        Cap sent = target;
        std::optional<typename Solver::Result> result;
        if (
            !target_infeasible &&
            target != std::numeric_limits<Cap>::max()
        ) {
            result = solver.min_cost_st_flow(s, t, target);
        }
        if (!result) {
            MaxFlow<Cap> feasible(_n);
            feasible.reserve_edges(int(arcs.size()));
            for (const auto& arc : arcs) {
                auto [from, idx] = _pos[arc.edge];
                const auto& e = _g[from][idx];
                const auto& reverse = _g[e.to][e.rev];
                if (arc.reverse) {
                    feasible.add_edge(e.to, from, reverse.cap);
                } else {
                    feasible.add_edge(from, e.to, e.cap);
                }
            }
            sent = feasible.max_flow(s, t, target);
            if (sent == Cap(0)) return {Cap(0), Cost(0)};
            result = solver.min_cost_st_flow(s, t, sent);
        }
        assert(result.has_value());
        for (int i = 0; i < int(arcs.size()); i++) {
            auto [from, idx] = _pos[arcs[i].edge];
            auto& e = _g[from][idx];
            auto& reverse = _g[e.to][e.rev];
            Cap amount = result->flow(i);
            if (arcs[i].reverse) {
                reverse.cap -= amount;
                e.cap += amount;
            } else {
                e.cap -= amount;
                reverse.cap += amount;
            }
        }
        _has_flow = true;
        return {sent, result->cost};
    }

    void init_potential(int s, std::vector<Cost>& potential, Cost cost_inf) const {
        if (!_has_negative_cost && !_has_flow) {
            potential.assign(_n, Cost(0));
            return;
        }
        potential.assign(_n, cost_inf);
        potential[s] = Cost(0);
        for (int iter = 0; iter < _n - 1; iter++) {
            bool updated = false;
            for (int v = 0; v < _n; v++) {
                if (potential[v] == cost_inf) continue;
                for (const auto& e : _g[v]) {
                    if (e.cap == Cap(0)) continue;
                    Cost nd = potential[v] + e.cost;
                    if (nd < potential[e.to]) {
                        potential[e.to] = nd;
                        updated = true;
                    }
                }
            }
            if (!updated) break;
        }
        for (int v = 0; v < _n; v++) {
            if (potential[v] == cost_inf) potential[v] = Cost(0);
        }
    }

   public:
    MinCostFlow() : MinCostFlow(0) {}

    explicit MinCostFlow(int n)
        : _n(n), _g(n), _has_negative_cost(false), _has_flow(false) {
        assert(0 <= n);
    }

    int size() const {
        return _n;
    }

    int edge_count() const {
        return int(_pos.size());
    }

    void reserve_edges(int edge_count) {
        assert(0 <= edge_count);
        _pos.reserve(edge_count);
        if (_n == 0 || edge_count == 0 ||
            2 * std::size_t(edge_count) < std::size_t(_n)) {
            return;
        }
        const std::size_t average_degree =
            (3 * std::size_t(edge_count) + std::size_t(_n) - 1)
            / std::size_t(_n);
        for (auto& edges : _g) edges.reserve(average_degree);
    }

    void reserve_edges(int edge_count, const std::vector<int>& degrees) {
        assert(0 <= edge_count);
        assert(int(degrees.size()) == _n);
        _pos.reserve(edge_count);
        for (int v = 0; v < _n; v++) {
            assert(0 <= degrees[v]);
            _g[v].reserve(degrees[v]);
        }
    }

    int add_edge(int from, int to, Cap cap, Cost cost) {
        assert(0 <= from && from < _n);
        assert(0 <= to && to < _n);
        assert(Cap(0) <= cap);
        _has_negative_cost = _has_negative_cost || cost < Cost(0);
        int id = int(_pos.size());
        int from_id = int(_g[from].size());
        int to_id = int(_g[to].size());
        if (from == to) to_id++;
        _pos.emplace_back(from, from_id);
        _g[from].push_back(InternalEdge{to, to_id, cap, cost});
        _g[to].push_back(InternalEdge{from, from_id, Cap(0), -cost});
        return id;
    }

    Edge get_edge(int i) const {
        assert(0 <= i && i < int(_pos.size()));
        auto [from, idx] = _pos[i];
        const auto& e = _g[from][idx];
        const auto& re = _g[e.to][e.rev];
        return Edge{from, e.to, e.cap + re.cap, re.cap, e.cost};
    }

    std::vector<Edge> edges() const {
        std::vector<Edge> result;
        result.reserve(_pos.size());
        for (int i = 0; i < int(_pos.size()); i++) result.push_back(get_edge(i));
        return result;
    }

    std::pair<Cap, Cost> flow(int s, int t) {
        return flow(s, t, std::numeric_limits<Cap>::max());
    }

    std::pair<Cap, Cost> flow(int s, int t, Cap flow_limit) {
        assert(0 <= s && s < _n);
        assert(0 <= t && t < _n);
        assert(s != t);
        assert(Cap(0) <= flow_limit);
        if (flow_limit == Cap(0)) return {Cap(0), Cost(0)};
        if constexpr (
            std::numeric_limits<Cap>::is_integer &&
            std::numeric_limits<Cap>::is_signed &&
            std::numeric_limits<Cost>::is_signed
        ) {
            if (use_network_simplex(s, t, flow_limit)) {
                return network_simplex_flow(s, t, flow_limit);
            }
        }
        auto result = slope(s, t, flow_limit);
        return result.back();
    }

    std::vector<std::pair<Cap, Cost>> slope(int s, int t) {
        return slope(s, t, std::numeric_limits<Cap>::max());
    }

    std::vector<std::pair<Cap, Cost>> slope(int s, int t, Cap flow_limit) {
        assert(0 <= s && s < _n);
        assert(0 <= t && t < _n);
        assert(s != t);
        assert(Cap(0) <= flow_limit);

        const Cost cost_inf = std::numeric_limits<Cost>::max() / Cost(4);
        std::vector<Cost> potential, dist(_n);
        std::vector<int> prev_v(_n), prev_e(_n);
        std::vector<int> settled;
        settled.reserve(_n);
        typename HeapSelector<Cost>::Type que;
        init_potential(s, potential, cost_inf);

        std::vector<std::pair<Cap, Cost>> result;
        result.emplace_back(Cap(0), Cost(0));
        Cap flow = 0;
        Cost cost = 0;

        while (flow < flow_limit) {
            std::fill(dist.begin(), dist.end(), cost_inf);
            dist[s] = Cost(0);
            settled.clear();
            que.clear();
            que.push(Cost(0), s);

            while (!que.empty()) {
                auto [d, v] = que.pop();
                if (dist[v] != d) continue;
                settled.push_back(v);
                if (v == t) break;
                for (int i = 0; i < int(_g[v].size()); i++) {
                    const auto& e = _g[v][i];
                    if (e.cap == Cap(0)) continue;
                    Cost nd = d + e.cost + potential[v] - potential[e.to];
                    if (nd >= dist[e.to]) continue;
                    dist[e.to] = nd;
                    prev_v[e.to] = v;
                    prev_e[e.to] = i;
                    que.push(nd, e.to);
                }
            }

            if (dist[t] == cost_inf) break;
            for (int v : settled) {
                potential[v] += dist[v] - dist[t];
            }

            Cap add = flow_limit - flow;
            for (int v = t; v != s; v = prev_v[v]) {
                add = std::min(add, _g[prev_v[v]][prev_e[v]].cap);
            }
            Cost path_cost = potential[t] - potential[s];
            for (int v = t; v != s; v = prev_v[v]) {
                auto& e = _g[prev_v[v]][prev_e[v]];
                e.cap -= add;
                _g[e.to][e.rev].cap += add;
            }

            flow += add;
            cost += Cost(add) * path_cost;
            result.emplace_back(flow, cost);
        }

        _has_flow = _has_flow || flow != Cap(0);
        return result;
    }
};

}  // namespace flow
}  // namespace m1une


#line 1 "utilities/fast_io.hpp"



#line 6 "utilities/fast_io.hpp"
#include <cerrno>
#include <charconv>
#line 9 "utilities/fast_io.hpp"
#include <cstdio>
#include <cstdlib>
#include <cstdint>
#include <cstring>
#include <iterator>
#include <string>
#include <sys/stat.h>
#line 18 "utilities/fast_io.hpp"
#include <unistd.h>
#line 20 "utilities/fast_io.hpp"

namespace m1une {
namespace utilities {

struct FastOutput;

namespace internal {

// Shared with the convenience helpers in template.hpp.
inline FastOutput* standard_output_instance = nullptr;

// Detect std::begin(x), std::end(x).
template <class T, class = void>
struct is_range : std::false_type {};

template <class T>
struct is_range<T, std::void_t<
    decltype(std::begin(std::declval<T&>())),
    decltype(std::end(std::declval<T&>()))
>> : std::true_type {};

template <class T>
inline constexpr bool is_range_v = is_range<T>::value;

template <class T>
using range_reference_t = decltype(*std::begin(std::declval<T&>()));

template <class T>
using range_value_t = std::remove_cv_t<std::remove_reference_t<range_reference_t<T>>>;

template <class T, class = void>
struct range_stored_value {
    using type = range_value_t<T>;
};

template <class T>
struct range_stored_value<T, std::void_t<typename std::remove_cv_t<std::remove_reference_t<T>>::value_type>> {
    using type = typename std::remove_cv_t<std::remove_reference_t<T>>::value_type;
};

template <class T>
using range_stored_value_t = typename range_stored_value<T>::type;

// Treat strings and C strings as scalar output objects, not as ranges.
template <class T>
struct is_char_array : std::false_type {};

template <class T, std::size_t N>
struct is_char_array<T[N]>
    : std::bool_constant<std::is_same_v<std::remove_cv_t<T>, char>> {};

template <class T>
struct is_string_like
    : std::bool_constant<
          std::is_same_v<std::decay_t<T>, std::string>
          || std::is_same_v<std::decay_t<T>, const char*>
          || std::is_same_v<std::decay_t<T>, char*>
          || is_char_array<std::remove_reference_t<T>>::value
      > {};

template <class T>
inline constexpr bool is_string_like_v = is_string_like<T>::value;

// ModInt-like type: x.val() is printable, and x can be assigned from long long.
template <class T, class = void>
struct has_val_method : std::false_type {};

template <class T>
struct has_val_method<T, std::void_t<decltype(std::declval<const T&>().val())>>
    : std::true_type {};

template <class T>
inline constexpr bool has_val_method_v = has_val_method<T>::value;

template <class T, class = void>
struct has_static_mod_raw : std::false_type {};

template <class T>
struct has_static_mod_raw<
    T, std::void_t<decltype(T::mod()), decltype(T::raw(std::declval<uint32_t>()))>>
    : std::true_type {};

template <class T>
inline constexpr bool has_static_mod_raw_v = has_static_mod_raw<T>::value;

// libstdc++ before GCC 16 does not classify __int128 as an integral type in
// strict ISO modes such as -std=c++23. Keep the fast-I/O interface independent
// of that implementation detail.
template <class T>
inline constexpr bool is_integral_v =
    std::is_integral_v<T>
    || std::is_same_v<std::remove_cv_t<T>, __int128_t>
    || std::is_same_v<std::remove_cv_t<T>, __uint128_t>;

template <class T>
inline constexpr bool is_signed_v =
    std::is_signed_v<T>
    || std::is_same_v<std::remove_cv_t<T>, __int128_t>;

template <class T>
struct make_unsigned {
    using type = std::make_unsigned_t<T>;
};

template <>
struct make_unsigned<__int128_t> {
    using type = __uint128_t;
};

template <>
struct make_unsigned<__uint128_t> {
    using type = __uint128_t;
};

template <class T>
using make_unsigned_t = typename make_unsigned<std::remove_cv_t<T>>::type;

}  // namespace internal

struct FastInput {
    static constexpr int buffer_size = 1 << 20;

   private:
    std::FILE* _stream;
    char _buffer[buffer_size];
    int _position;
    int _length;
    int _file_descriptor;
    bool _streaming;

    bool refill() {
        _position = 0;
        if (_streaming) {
            ssize_t length;
            do {
                length = ::read(_file_descriptor, _buffer, buffer_size);
            } while (length < 0 && errno == EINTR);
            if (length <= 0) {
                _length = 0;
                return false;
            }
            _length = int(length);
        } else {
            _length = int(std::fread(_buffer, 1, buffer_size, _stream));
        }
        return _length != 0;
    }

    template <class T>
    bool read_integer_from_stream(T& value) {
        if (!skip_spaces()) return false;
        int c = read_char_raw();

        bool negative = false;
        if (c == '-') {
            negative = true;
            c = read_char_raw();
        }

        if constexpr (internal::is_signed_v<T>) {
            T result = 0;
            while ('0' <= c && c <= '9') {
                result = negative ? result * 10 - (c - '0')
                                  : result * 10 + (c - '0');
                c = read_char_raw();
            }
            value = result;
        } else {
            T result = 0;
            while ('0' <= c && c <= '9') {
                result = result * 10 + T(c - '0');
                c = read_char_raw();
            }
            value = negative ? T(0) - result : result;
        }
        return true;
    }

    bool prepare_number() {
        if (_length - _position >= 64) return true;
        const int remaining = _length - _position;
        if (remaining > 0) std::memmove(_buffer, _buffer + _position, remaining);
        const int added = int(std::fread(_buffer + remaining, 1, buffer_size - remaining, _stream));
        _position = 0;
        _length = remaining + added;
        if (_length < buffer_size) _buffer[_length] = '\0';
        return _length != 0;
    }

   public:
    explicit FastInput(std::FILE* stream = stdin)
        : _stream(stream),
          _position(0),
          _length(0),
          _file_descriptor(::fileno(stream)),
          _streaming([&] {
              struct stat status;
              return _file_descriptor >= 0
                     && ::fstat(_file_descriptor, &status) == 0
                     && !S_ISREG(status.st_mode);
          }()) {}

    FastInput(const FastInput&) = delete;
    FastInput& operator=(const FastInput&) = delete;

    int read_char_raw() {
        if (_position == _length && !refill()) return EOF;
        return _buffer[_position++];
    }

    bool skip_spaces() {
        int c = read_char_raw();
        while (c != EOF && c <= ' ') c = read_char_raw();
        if (c == EOF) return false;
        --_position;
        return true;
    }

    bool read(char& value) {
        if (!skip_spaces()) return false;
        value = char(read_char_raw());
        return true;
    }

    bool read(std::string& value) {
        if (!skip_spaces()) return false;
        value.clear();
        while (true) {
            const int begin = _position;
            while (_position < _length &&
                   static_cast<unsigned char>(_buffer[_position]) > ' ') {
                ++_position;
            }
            value.append(_buffer + begin, _position - begin);
            if (_position < _length) {
                ++_position;
                return true;
            }
            if (!refill()) return true;
        }
    }

    bool read(bool& value) {
        int x;
        if (!read(x)) return false;
        value = x != 0;
        return true;
    }

    template <class T>
    std::enable_if_t<
        internal::is_integral_v<T>
            && !std::is_same_v<std::remove_cv_t<T>, bool>
            && !std::is_same_v<std::remove_cv_t<T>, char>,
        bool
    >
    read(T& value) {
        if (_streaming) return read_integer_from_stream(value);
        if (!prepare_number()) return false;
        int c = static_cast<unsigned char>(_buffer[_position++]);
        while (c <= ' ') c = static_cast<unsigned char>(_buffer[_position++]);

        bool negative = false;
        if (c == '-') {
            negative = true;
            c = static_cast<unsigned char>(_buffer[_position++]);
        }

        if constexpr (internal::is_signed_v<T>) {
            T result = 0;
            while ('0' <= c && c <= '9') {
                const int first = c - '0';
                const int second = static_cast<unsigned char>(_buffer[_position]) - '0';
                if (0 <= second && second <= 9) {
                    result = negative ? result * 100 - (first * 10 + second)
                                      : result * 100 + (first * 10 + second);
                    ++_position;
                } else {
                    result = negative ? result * 10 - first : result * 10 + first;
                }
                c = static_cast<unsigned char>(_buffer[_position++]);
            }
            value = result;
        } else {
            T result = 0;
            while ('0' <= c && c <= '9') {
                const unsigned first = unsigned(c - '0');
                const int second = static_cast<unsigned char>(_buffer[_position]) - '0';
                if (0 <= second && second <= 9) {
                    result = result * 100 + T(first * 10 + unsigned(second));
                    ++_position;
                } else {
                    result = result * 10 + T(first);
                }
                c = static_cast<unsigned char>(_buffer[_position++]);
            }
            value = negative ? T(0) - result : result;
        }
        if (_position > _length) _position = _length;
        return true;
    }

    template <class T>
    std::enable_if_t<std::is_floating_point_v<T>, bool>
    read(T& value) {
        if (!skip_spaces()) return false;
        int c = read_char_raw();
        bool negative = false;
        if (c == '-' || c == '+') {
            negative = c == '-';
            c = read_char_raw();
        }

        long double result = 0;
        while ('0' <= c && c <= '9') {
            result = result * 10 + (c - '0');
            c = read_char_raw();
        }
        if (c == '.') {
            long double place = 0.1L;
            c = read_char_raw();
            while ('0' <= c && c <= '9') {
                result += (c - '0') * place;
                place *= 0.1L;
                c = read_char_raw();
            }
        }
        if (c == 'e' || c == 'E') {
            c = read_char_raw();
            bool exponent_negative = false;
            if (c == '-' || c == '+') {
                exponent_negative = c == '-';
                c = read_char_raw();
            }
            int exponent = 0;
            while ('0' <= c && c <= '9') {
                exponent = exponent * 10 + (c - '0');
                c = read_char_raw();
            }
            long double scale = 1;
            long double power = 10;
            while (exponent > 0) {
                if (exponent & 1) scale *= power;
                power *= power;
                exponent >>= 1;
            }
            result = exponent_negative ? result / scale : result * scale;
        }
        value = static_cast<T>(negative ? -result : result);
        return true;
    }

    template <class T>
    std::enable_if_t<
        internal::has_val_method_v<T>
            && !internal::is_integral_v<T>
            && !internal::is_range_v<T>,
        bool
    >
    read(T& value) {
        long long x;
        if (!read(x)) return false;
        if constexpr (internal::has_static_mod_raw_v<T>) {
            if (x >= 0 && uint64_t(x) < uint64_t(T::mod())) {
                value = T::raw(uint32_t(x));
            } else {
                value = T(x);
            }
        } else {
            value = T(x);
        }
        return true;
    }

    template <class First, class Second>
    bool read(std::pair<First, Second>& value) {
        if (!read(value.first)) return false;
        return read(value.second);
    }

    template <class Range>
    std::enable_if_t<
        internal::is_range_v<Range>
            && !internal::is_string_like_v<Range>,
        bool
    >
    read(Range& range) {
        using StoredValue = internal::range_stored_value_t<Range>;
        constexpr bool nested = internal::is_range_v<StoredValue>
                                && !internal::is_string_like_v<StoredValue>;

        for (auto&& value : range) {
            if constexpr (std::is_same_v<StoredValue, bool> && !nested) {
                bool x;
                if (!read(x)) return false;
                value = x;
            } else {
                if (!read(value)) return false;
            }
        }
        return true;
    }

    template <class First, class Second, class... Rest>
    bool read(First& first, Second& second, Rest&... rest) {
        if (!read(first)) return false;
        return read(second, rest...);
    }

    template <class T>
    FastInput& operator>>(T& value) {
        if (!read(value)) std::abort();
        return *this;
    }
};

struct FastOutput {
    static constexpr int buffer_size = 1 << 20;

   private:
    inline static const auto digit_quads = [] {
        std::array<char, 40000> result{};
        for (int i = 0; i < 10000; i++) {
            int value = i;
            for (int j = 3; j >= 0; j--) {
                result[4 * i + j] = char('0' + value % 10);
                value /= 10;
            }
        }
        return result;
    }();

    std::FILE* _stream;
    char _buffer[buffer_size];
    int _position;
    int _precision;
    std::chars_format _float_format;
    char _range_separator;
    std::string* _capture = nullptr;

    template <class T>
    std::string format_cell(const T& value) {
        std::string result;
        struct CaptureGuard {
            std::string*& target;
            std::string* previous;
            ~CaptureGuard() { target = previous; }
        } guard{_capture, _capture};
        _capture = &result;
        write(value);
        return result;
    }

    template <class Matrix>
    void write_aligned_matrix(const Matrix& matrix) {
        std::vector<std::vector<std::string>> rows;
        std::vector<std::size_t> widths;
        for (const auto& row : matrix) {
            auto& cells = rows.emplace_back();
            std::size_t column = 0;
            for (const auto& value : row) {
                cells.push_back(format_cell(value));
                if (column == widths.size()) widths.push_back(0);
                widths[column] = std::max(widths[column], cells.back().size());
                ++column;
            }
        }
        bool first = true;
        for (const auto& row : rows) {
            if (!first) write_char('\n');
            first = false;
            for (std::size_t column = 0; column < row.size(); ++column) {
                if (column != 0) write_char(_range_separator);
                for (std::size_t padding = row[column].size();
                     padding < widths[column]; ++padding) {
                    write_char(' ');
                }
                write(row[column]);
            }
        }
    }

   public:
    explicit FastOutput(std::FILE* stream = stdout)
        : _stream(stream),
          _position(0),
          _precision(6),
          _float_format(std::chars_format::general),
          _range_separator(' ') {
        if (_stream == stdout
            && internal::standard_output_instance == nullptr) {
            internal::standard_output_instance = this;
        }
    }

    FastOutput(const FastOutput&) = delete;
    FastOutput& operator=(const FastOutput&) = delete;

    ~FastOutput() {
        flush();
        if (internal::standard_output_instance == this) {
            internal::standard_output_instance = nullptr;
        }
    }

    void flush() {
        if (_position != 0) {
            std::fwrite(_buffer, 1, _position, _stream);
            _position = 0;
        }
        std::fflush(_stream);
    }

    void write_char(char c) {
        if (_capture != nullptr) {
            _capture->push_back(c);
            return;
        }
        if (_position == buffer_size) flush();
        _buffer[_position++] = c;
    }

    void write(const char* s) {
        while (*s != '\0') write_char(*s++);
    }

    void write(const std::string& s) {
        if (_capture != nullptr) {
            _capture->append(s);
            return;
        }
        std::size_t position = 0;
        while (position < s.size()) {
            if (_position == buffer_size) flush();
            const std::size_t copied =
                std::min<std::size_t>(buffer_size - _position, s.size() - position);
            std::memcpy(_buffer + _position, s.data() + position, copied);
            _position += int(copied);
            position += copied;
        }
    }

    void write(char c) {
        write_char(c);
    }

    void write(bool value) {
        write_char(value ? '1' : '0');
    }

    template <class T>
    std::enable_if_t<std::is_floating_point_v<T>>
    write(T value) {
        char digits[128];
        auto [end, error] = std::to_chars(
            digits,
            digits + sizeof(digits),
            value,
            _float_format,
            _precision
        );
        if (error != std::errc()) std::abort();
        for (const char* pointer = digits; pointer != end; pointer++) {
            write_char(*pointer);
        }
    }

    template <class T>
    std::enable_if_t<
        internal::is_integral_v<T>
            && !std::is_same_v<std::remove_cv_t<T>, bool>
            && !std::is_same_v<std::remove_cv_t<T>, char>
    >
    write(T value) {
        using Raw = std::remove_cv_t<T>;
        using Unsigned = internal::make_unsigned_t<Raw>;

        Unsigned magnitude;
        if constexpr (internal::is_signed_v<Raw>) {
            if (value < 0) {
                write_char('-');
                magnitude = Unsigned(0) - Unsigned(value);
            } else {
                magnitude = Unsigned(value);
            }
        } else {
            magnitude = value;
        }

        if (magnitude == 0) {
            write_char('0');
            return;
        }

        unsigned chunks[16];
        int count = 0;
        while (magnitude >= 10000) {
            const Unsigned quotient = magnitude / 10000;
            chunks[count++] = unsigned(magnitude - quotient * 10000);
            magnitude = quotient;
        }
        if (_capture == nullptr && _position > buffer_size - 64) flush();
        char captured[64];
        char* const begin = _capture != nullptr ? captured : _buffer + _position;
        char* destination = begin;
        const unsigned leading = unsigned(magnitude);
        const char* first = digit_quads.data() + 4 * leading;
        int skip = leading < 10 ? 3 : leading < 100 ? 2 : leading < 1000 ? 1 : 0;
        for (; skip < 4; skip++) *destination++ = first[skip];
        while (count--) {
            const char* digits = digit_quads.data() + 4 * chunks[count];
            std::memcpy(destination, digits, 4);
            destination += 4;
        }
        if (_capture != nullptr) {
            _capture->append(begin, destination - begin);
        } else {
            _position += int(destination - begin);
        }
    }

    template <class T>
    std::enable_if_t<
        internal::has_val_method_v<T>
            && !internal::is_integral_v<T>
            && !internal::is_range_v<T>
    >
    write(const T& value) {
        write(value.val());
    }

    template <class First, class Second>
    void write(const std::pair<First, Second>& value) {
        write(value.first);
        write_char(' ');
        write(value.second);
    }

    template <class Range>
    std::enable_if_t<
        internal::is_range_v<Range>
            && !internal::is_string_like_v<Range>
    >
    write(const Range& range) {
        using StoredValue = internal::range_stored_value_t<const Range>;
        constexpr bool nested = internal::is_range_v<StoredValue>
                                && !internal::is_string_like_v<StoredValue>;

        bool first = true;
        for (const auto& value : range) {
            if (!first) write_char(nested ? '\n' : _range_separator);
            first = false;
            if constexpr (std::is_same_v<StoredValue, bool> && !nested) {
                write(static_cast<bool>(value));
            } else {
                write(value);
            }
        }
    }

    template <class First, class... Rest>
    void print(const First& first, const Rest&... rest) {
        write(first);
        ((write_char(' '), write(rest)), ...);
    }

    void println() {
        write_char('\n');
    }

    void set_precision(int precision) {
        _precision = precision;
    }

    void set_fixed(int precision = 6) {
        _float_format = std::chars_format::fixed;
        _precision = precision;
    }

    void set_general(int precision = 6) {
        _float_format = std::chars_format::general;
        _precision = precision;
    }

    void set_range_separator(char separator) {
        _range_separator = separator;
    }

    template <class Matrix>
    void write_aligned(const Matrix& matrix) {
        using Row = internal::range_stored_value_t<const Matrix>;
        using Cell = internal::range_stored_value_t<const Row>;
        static_assert(internal::is_range_v<Row> && !internal::is_string_like_v<Row>,
                      "write_aligned requires a two-dimensional range");
        static_assert(!internal::is_range_v<Cell> || internal::is_string_like_v<Cell>,
                      "write_aligned requires scalar cells");
        write_aligned_matrix(matrix);
    }

    template <class Matrix>
    void println_aligned(const Matrix& matrix) {
        write_aligned(matrix);
        write_char('\n');
    }

    template <class... Args>
    void println(const Args&... args) {
        print(args...);
        write_char('\n');
    }

    template <class T>
    FastOutput& operator<<(const T& value) {
        write(value);
        return *this;
    }
};

}  // namespace utilities
}  // namespace m1une


#line 5 "verify/graph/flow/min_cost_flow.test.cpp"

int main() {
    m1une::utilities::FastInput input;
    m1une::utilities::FastOutput output;

    int vertex_count, edge_count;
    long long required_flow;
    input >> vertex_count >> edge_count >> required_flow;
    m1une::flow::MinCostFlow<long long, long long> flow(vertex_count);
    flow.reserve_edges(edge_count);
    for (int edge = 0; edge < edge_count; edge++) {
        int from, to;
        long long capacity, cost;
        input >> from >> to >> capacity >> cost;
        flow.add_edge(from, to, capacity, cost);
    }
    auto [sent, cost] = flow.flow(0, vertex_count - 1, required_flow);
    output << (sent == required_flow ? cost : -1) << '\n';
}

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