匈牙利算法

首先，他是一个求二分图最大匹配的算法。

二分图

有两个独立的点集U,V，二分图的任意一个边的两个端点分别来自U和V；所谓最大匹配就是找到最大匹配的个数。

算法思想

：就是找可增广轨(若图中有一轨，其边交替地在对集 内外出现，则称此轨为 的交错轨，交错轨的起止顶点都未被许配时，此交错轨称为可增广轨）；找到之后，把可增广轨上在 外的边纳入对集，把 内的边从对集中删除，则被许配的顶点数增加2，对集中的“对儿”增加一个。直到找不到可增广轨。

matlab代码

function [Matching,Cost] = Edmonds(a)
Matching = zeros(size(a));
num_y = sum(~isinf(a),1);
num_x = sum(~isinf(a),2);
x_con = find(num_x~=0);
y_con = find(num_y~=0);
P_size = max(length(x_con),length(y_con));
P_cond = zeros(P_size);
P_cond(1:length(x_con),1:length(y_con)) = a(x_con,y_con);
if isempty(P_cond)
Cost = 0;
return
end
Edge = P_cond;
Edge(P_cond~=Inf) = 0;
cnum = min_line_cover(Edge);
Pmax = max(max(P_cond(P_cond~=Inf)));
P_size = length(P_cond)+cnum;
P_cond = ones(P_size)*Pmax;
P_cond(1:length(x_con),1:length(y_con)) = a(x_con,y_con);
exit_flag = 1;
stepnum = 1;
while exit_flag
switch stepnum
case 1
[P_cond,stepnum] = step1(P_cond);
case 2
[r_cov,c_cov,M,stepnum] = step2(P_cond);
case 3
[c_cov,stepnum] = step3(M,P_size);
case 4
[M,r_cov,c_cov,Z_r,Z_c,stepnum] = step4(P_cond,r_cov,c_cov,M);
case 5
[M,r_cov,c_cov,stepnum] = step5(M,Z_r,Z_c,r_cov,c_cov);
case 6
[P_cond,stepnum] = step6(P_cond,r_cov,c_cov);
case 7
exit_flag = 0;
end
end
Matching(x_con,y_con) = M(1:length(x_con),1:length(y_con));
Cost = sum(sum(a(Matching==1)));
function [P_cond,stepnum] = step1(P_cond)
P_size = length(P_cond);
for ii = 1:P_size
rmin = min(P_cond(ii,:));
P_cond(ii,:) = P_cond(ii,:)-rmin;
end
stepnum = 2;
function [r_cov,c_cov,M,stepnum] = step2(P_cond)
P_size = length(P_cond);
r_cov = zeros(P_size,1);
c_cov = zeros(P_size,1);
M = zeros(P_size);
for ii = 1:P_size
for jj = 1:P_size
if P_cond(ii,jj) == 0 && r_cov(ii) == 0 && c_cov(jj) == 0
M(ii,jj) = 1;
r_cov(ii) = 1;
c_cov(jj) = 1;
end
end
end
r_cov = zeros(P_size,1);  % A vector that shows if a row is covered
c_cov = zeros(P_size,1);  % A vector that shows if a column is covered
stepnum = 3;
function [c_cov,stepnum] = step3(M,P_size)
c_cov = sum(M,1);
if sum(c_cov) == P_size
stepnum = 7;
else
stepnum = 4;
end
function [M,r_cov,c_cov,Z_r,Z_c,stepnum] = step4(P_cond,r_cov,c_cov,M)
P_size = length(P_cond);
zflag = 1;
while zflag
row = 0; col = 0; exit_flag = 1;
ii = 1; jj = 1;
while exit_flag
if P_cond(ii,jj) == 0 && r_cov(ii) == 0 && c_cov(jj) == 0
row = ii;
col = jj;
exit_flag = 0;
end
jj = jj + 1;
if jj > P_size; jj = 1; ii = ii+1; end
if ii > P_size; exit_flag = 0; end
end
if row == 0
stepnum = 6;
zflag = 0;
Z_r = 0;
Z_c = 0;
else
M(row,col) = 2;
if sum(find(M(row,:)==1)) ~= 0
r_cov(row) = 1;
zcol = find(M(row,:)==1);
c_cov(zcol) = 0;
else
stepnum = 5;
zflag = 0;
Z_r = row;
Z_c = col;
end
end
end
function [M,r_cov,c_cov,stepnum] = step5(M,Z_r,Z_c,r_cov,c_cov)
zflag = 1;
ii = 1;
while zflag
rindex = find(M(:,Z_c(ii))==1);
if rindex > 0
ii = ii+1;
Z_r(ii,1) = rindex;
Z_c(ii,1) = Z_c(ii-1);
else
zflag = 0;
end
if zflag == 1;
cindex = find(M(Z_r(ii),:)==2);
ii = ii+1;
Z_r(ii,1) = Z_r(ii-1);
Z_c(ii,1) = cindex;
end
end
for ii = 1:length(Z_r)
if M(Z_r(ii),Z_c(ii)) == 1
M(Z_r(ii),Z_c(ii)) = 0;
else
M(Z_r(ii),Z_c(ii)) = 1;
end
end
r_cov = r_cov.*0;
c_cov = c_cov.*0;
M(M==2) = 0;
stepnum = 3;
function [P_cond,stepnum] = step6(P_cond,r_cov,c_cov)
a = find(r_cov == 0);
b = find(c_cov == 0);
minval = min(min(P_cond(a,b)));
P_cond(find(r_cov == 1),:) = P_cond(find(r_cov == 1),:) + minval;
P_cond(:,find(c_cov == 0)) = P_cond(:,find(c_cov == 0)) - minval;
stepnum = 4;
function cnum = min_line_cover(Edge)
[r_cov,c_cov,M,stepnum] = step2(Edge);
[c_cov,stepnum] = step3(M,length(Edge));
[M,r_cov,c_cov,Z_r,Z_c,stepnum] = step4(Edge,r_cov,c_cov,M);
cnum = length(Edge)-sum(r_cov)-sum(c_cov);

其中matching为匹配矩阵

cost为权值最小。