MATLAB实现最小费用最大流算法代码

以下是一个MATLAB实现最小费用最大流算法的代码示例，使用了Ford-Fulkerson算法和Bellman-Ford算法：

function [max_flow, min_cost] = min_cost_max_flow(capacity, cost, source, sink)
% capacity: 有向图的容量矩阵
% cost: 有向图的费用矩阵
% source: 源点
% sink: 汇点
% max_flow: 最大流量
% min_cost: 最小费用

n = size(capacity, 1); % 节点数
flow = zeros(n); % 流量矩阵
residual_capacity = capacity; % 剩余容量矩阵
residual_cost = cost; % 剩余费用矩阵
max_flow = 0; % 最大流量
min_cost = 0; % 最小费用

while true
    % 使用Ford-Fulkerson算法计算增广路径
    [aug_path, aug_capacity] = find_aug_path(residual_capacity, source, sink);
    
    if isempty(aug_path)
        % 没有增广路径，计算结束
        break;
    end
    
    % 计算增广路径的费用
    aug_cost = compute_aug_cost(aug_path, aug_capacity, residual_cost);
    
    % 更新流量和剩余容量
    max_flow = max_flow + aug_capacity;
    for i = 1:length(aug_path)-1
        u = aug_path(i);
        v = aug_path(i+1);
        flow(u, v) = flow(u, v) + aug_capacity;
        flow(v, u) = flow(v, u) - aug_capacity;
        residual_capacity(u, v) = residual_capacity(u, v) - aug_capacity;
        residual_capacity(v, u) = residual_capacity(v, u) + aug_capacity;
    end
    
    % 更新剩余费用
    residual_cost = update_residual_cost(residual_capacity, cost, flow);
    
    % 更新总费用
    min_cost = min_cost + aug_cost;
end

end

function [aug_path, aug_capacity] = find_aug_path(residual_capacity, source, sink)
% 使用Bellman-Ford算法查找增广路径
n = size(residual_capacity, 1);
dist = Inf(1, n);
dist(source) = 0;
prev = zeros(1, n);
queue = [source];
while ~isempty(queue)
    u = queue(1);
    queue(1) = [];
    for v = 1:n
        if residual_capacity(u, v) > 0 && dist(u) + residual_capacity(u, v) < dist(v)
            dist(v) = dist(u) + residual_capacity(u, v);
            prev(v) = u;
            queue(end+1) = v;
        end
    end
end
if prev(sink) == 0
    % 没有增广路径
    aug_path = [];
    aug_capacity = 0;
else
    % 计算增广路径和容量
    aug_path = [sink];
    aug_capacity = Inf;
    while aug_path(end) ~= source
        u = prev(aug_path(end));
        v = aug_path(end);
        aug_path(end+1) = u;
        aug_capacity = min(aug_capacity, residual_capacity(u, v));
    end
    aug_path = fliplr(aug_path);
end
end

function aug_cost = compute_aug_cost(aug_path, aug_capacity, residual_cost)
% 计算增广路径的费用
aug_cost = 0;
for i = 1:length(aug_path)-1
    u = aug_path(i);
    v = aug_path(i+1);
    aug_cost = aug_cost + residual_cost(u, v) * aug_capacity;
end
end

function residual_cost = update_residual_cost(residual_capacity, cost, flow)
% 更新剩余费用
residual_cost = cost;
for u = 1:size(residual_capacity, 1)
    for v = 1:size(residual_capacity, 2)
        if residual_capacity(u, v) > 0
            residual_cost(u, v) = cost(u, v) - flow(u, v);
        else
            residual_cost(u, v) = Inf;
        end
    end
end
end

注意，这个实现并不是最优的，仅供参考。在实际使用中，可能需要根据具体情况进行优化。