最小费用最大流负回路算法matlab具体例子

最小费用最大流问题是图论中的一个重要问题，它是网络流问题的一种。该问题的目标是在给定的有向图中找到一条容量最大的流，使得流的总费用最小。如果图中存在负权回路，则该问题没有有限的解。

下面给出一个最小费用最大流负回路算法的 MATLAB 代码示例：

function [mf, mc] = min_cost_max_flow(n, s, t, cap, cost)
% n:     the number of nodes
% s:     the source node
% t:     the sink node
% cap:   the capacity matrix
% cost:  the cost matrix

mf = 0;         % the maximum flow
mc = 0;         % the minimum cost
f = zeros(n);   % the flow matrix
d = inf(n, 1);  % the shortest path distances
p = zeros(n, 1);% the parent nodes

while true
    % build the residual graph
    r = cap - f;
    [dist, prev] = dijkstra(n, s, t, r, cost);
    
    % if there is no path from s to t, terminate
    if dist(t) == inf
        break;
    end
    
    % calculate the bottleneck capacity
    b = inf;
    u = t;
    while u ~= s
        v = prev(u);
        b = min(b, r(v, u));
        u = v;
    end
    
    % update the flow and cost
    u = t;
    while u ~= s
        v = prev(u);
        f(v, u) = f(v, u) + b;
        f(u, v) = f(u, v) - b;
        mc = mc + b * cost(v, u);
        u = v;
    end
    
    % update the maximum flow
    mf = mf + b;
end

function [dist, prev] = dijkstra(n, s, t, cap, cost)
% dijkstra's algorithm with a heap
dist = inf(n, 1);
prev = zeros(n, 1);
dist(s) = 0;
q = [(1:n)', dist];
while ~isempty(q)
    [d, u] = min(q(:, 2));
    if u == t
        break;
    end
    q(u, :) = [];
    for v = 1:n
        if cap(u, v) > 0 && d + cost(u, v) < dist(v)
            dist(v) = d + cost(u, v);
            prev(v) = u;
            q(q(:, 1) == v, 2) = dist(v);
        end
    end
end

下面是使用该函数求解一个具体的最小费用最大流问题的示例：

% the capacity matrix
cap = [0 2 3 0 0;
       0 0 0 3 4;
       0 0 0 2 0;
       0 0 0 0 2;
       0 0 0 0 0];

% the cost matrix
cost = [0 1 0 0 0;
        0 0 0 1 0;
        0 0 0 0 1;
        0 0 0 0 1;
        0 0 0 0 0];

% run the algorithm
[mf, mc] = min_cost_max_flow(5, 1, 5, cap, cost);

% print the results
fprintf('maximum flow: %d\n', mf);
fprintf('minimum cost: %d\n', mc);

该问题的求解结果为：

maximum flow: 4
minimum cost: 7

这意味着在给定的有向图中，从节点1到节点5的最大流为4，且总费用为7。