最小费用流matlab算法

最小费用流是一种在网络流中求解最小费用的问题，可以使用多种算法来解决。其中一种常见的算法是基于SPFA和Dijkstra的增广路算法。

以下是使用Matlab实现最小费用流的示例代码：

function [flow, cost] = min_cost_flow(capacity, cost, source, sink, flow_req)
% capacity: 网络中每条边的容量
% cost: 网络中每条边的费用
% source: 源点
% sink: 汇点
% flow_req: 每个节点需要的流量

n = length(capacity);
flow = zeros(n);
cost = zeros(n);

while true
    % 使用SPFA算法寻找增广路
    [dist, prev] = spfa(capacity, cost, source, sink);
    if dist(sink) == Inf
        break;
    end
    
    % 计算增广路上的最大流量
    path_flow = Inf;
    node = sink;
    while node ~= source
        path_flow = min(path_flow, capacity(prev(node), node) - flow(prev(node), node));
        node = prev(node);
    end
    
    % 更新流量和费用
    node = sink;
    while node ~= source
        flow(prev(node), node) = flow(prev(node), node) + path_flow;
        flow(node, prev(node)) = flow(node, prev(node)) - path_flow;
        cost(prev(node), node) = cost(prev(node), node) + path_flow * cost(prev(node), node);
        cost(node, prev(node)) = cost(node, prev(node)) - path_flow * cost(prev(node), node);
        node = prev(node);
    end
end

% 计算总流量和总费用
flow = sum(flow, 2);
cost = sum(cost .* flow, 2);

end

function [dist, prev] = spfa(capacity, cost, source, sink)
% 使用SPFA算法寻找增广路
n = length(capacity);
dist = Inf(n, 1);
prev = zeros(n, 1);
in_queue = false(n, 1);
queue = [];

dist(source) = 0;
in_queue(source) = true;
queue(end+1) = source;

while ~isempty(queue)
    node = queue(1);
    queue(1) = [];
    in_queue(node) = false;
    
    for neighbor = find(capacity(node, :) > 0)
        if capacity(node, neighbor) - flow(node, neighbor) > 0 && dist(node) + cost(node, neighbor) < dist(neighbor)
            dist(neighbor) = dist(node) + cost(node, neighbor);
            prev(neighbor) = node;
            if ~in_queue(neighbor)
                in_queue(neighbor) = true;
                queue(end+1) = neighbor;
            end
        end
    end
end

end