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  endMatching(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;  endfunction [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                  endendfunction [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。