匈牙利算法的理论依据-最优解定理

费用矩阵的一行(或列)的各个元素减去该行(或列)的最小元素所得到的新费用矩阵，与原费用矩阵对应的指派问题具有相同的最优解！

匈牙利算法的基本思路

• 通过行/列变换让费用矩阵的每行和每列都出现0
• 找出不同行不同列的n个0
• 这些0对应的指派就是最优指派

匈牙利算法的主要步骤

1. 步骤一：对费用矩阵先作行变换，再作列变换
   行变换：费用矩阵的每一行的各个元素分别减去该行的最小元素
   列变换：费用矩阵的每一列的各个元素分别减去该列的最小元素，有0的列则无需作列变换
2. 步骤二：在经过行变换和列变换的费用矩阵中寻找n个不同行不同列的0元素
   如找到，则这n个不同行不同列的0元素位置即对应最优指派
   否则，进行步骤三
   在步骤二中，一般使用标记法来寻找n个不同行不同列的0元素
   （1）依次检查新费用矩阵的各行，找出只有一个没有加标记的0元素的行，并将这个0元素加上标记“”，而与这个0元素在同一列的0元素全划去。
   （2）依次检查新费用矩阵的各列，找出只有一个没有加标记的0元素的列，并将这个0元素加上标记“”，而与这个0元素在同一行的0元素全划去。
3. 步骤三：对新费用矩阵进行调整
   （1）对每一个加了标记的0元素画一条横线或竖线，使得这些横线和竖线覆盖全部0元素
   （2）在这些横线和竖线没有经过的元素中找出最小的元素
   （3）未画横线的各行元素减去这个最小的数，画竖线的各列元素加上这个最小的数
   （4）重新在费用矩阵中找出n个不同行不同列的0元素，从而找出最优指派

更详细的内容见离散老师的课件2333

function [Matching,Cost] = Hungarian()
% 
% [MATCHING,COST] = Hungarian_New(WEIGHTS)
%
% A function for finding a minimum edge weight matching given a MxN Edge
% weight matrix WEIGHTS using the Hungarian Algorithm.
%
% An edge weight of Inf indicates that the pair of vertices given by its
% position have no adjacent edge.
%
% MATCHING return a MxN matrix with ones in the place of the matchings and
% zeros elsewhere.
% 
% COST returns the cost of the minimum matching

% Written by: Alex Melin 30 June 2006

Perf = [0 2 4 6 ; Inf 0 2 5; Inf 0 inf 4; inf inf inf 0];
 % Initialize Variables
 Matching = zeros(size(Perf));

% Condense the Performance Matrix by removing any unconnected vertices to
% increase the speed of the algorithm

  % Find the number in each column that are connected
    num_y = sum(~isinf(Perf),1);
  % Find the number in each row that are connected
    num_x = sum(~isinf(Perf),2);
    
  % Find the columns(vertices) and rows(vertices) that are isolated
    x_con = find(num_x~=0);
    y_con = find(num_y~=0);
    
  % Assemble Condensed Performance Matrix
    P_size = max(length(x_con),length(y_con));
    P_cond = zeros(P_size);
    P_cond(1:length(x_con),1:length(y_con)) = Perf(x_con,y_con);
    if isempty(P_cond)
      Cost = 0;
      return
    end

    % Ensure that a perfect matching exists
      % Calculate a form of the Edge Matrix
      Edge = P_cond;
      Edge(P_cond~=Inf) = 0;
      % Find the deficiency(CNUM) in the Edge Matrix
      cnum = min_line_cover(Edge);
    
      % Project additional vertices and edges so that a perfect matching
      % exists
      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)) = Perf(x_con,y_con);
   
%*************************************************
% MAIN PROGRAM: CONTROLS WHICH STEP IS EXECUTED
%*************************************************
  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

% Remove all the virtual satellites and targets and uncondense the
% Matching to the size of the original performance matrix.
Matching(x_con,y_con) = M(1:length(x_con),1:length(y_con));
Cost = sum(sum(Perf(Matching==1)));

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%   STEP 1: Find the smallest number of zeros in each row
%           and subtract that minimum from its row
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

function [P_cond,stepnum] = step1(P_cond)

  P_size = length(P_cond);
  
  % Loop throught each row
  for ii = 1:P_size
    rmin = min(P_cond(ii,:));
    P_cond(ii,:) = P_cond(ii,:)-rmin;
  end

  stepnum = 2;
  
%**************************************************************************  
%   STEP 2: Find a zero in P_cond. If there are no starred zeros in its
%           column or row start the zero. Repeat for each zero
%**************************************************************************

function [r_cov,c_cov,M,stepnum] = step2(P_cond)

% Define variables
  P_size = length(P_cond);
  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
  M = zeros(P_size);        % A mask that shows if a position is starred or primed
  
  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
  
% Re-initialize the cover vectors
  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;
  
%**************************************************************************
%   STEP 3: Cover each column with a starred zero. If all the columns are
%           covered then the matching is maximum
%**************************************************************************

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
  
%**************************************************************************
%   STEP 4: Find a noncovered zero and prime it.  If there is no starred
%           zero in the row containing this primed zero, Go to Step 5.  
%           Otherwise, cover this row and uncover the column containing 
%           the starred zero. Continue in this manner until there are no 
%           uncovered zeros left. Save the smallest uncovered value and 
%           Go to Step 6.
%**************************************************************************
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  
    % Find the first uncovered zero
      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 there are no uncovered zeros go to step 6
      if row == 0
        stepnum = 6;
        zflag = 0;
        Z_r = 0;
        Z_c = 0;
      else
        % Prime the uncovered zero
        M(row,col) = 2;
        % If there is a starred zero in that row
        % Cover the row and uncover the column containing the zero
          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
  
%**************************************************************************
% STEP 5: Construct a series of alternating primed and starred zeros as
%         follows.  Let Z0 represent the uncovered primed zero found in Step 4.
%         Let Z1 denote the starred zero in the column of Z0 (if any). 
%         Let Z2 denote the primed zero in the row of Z1 (there will always
%         be one).  Continue until the series terminates at a primed zero
%         that has no starred zero in its column.  Unstar each starred 
%         zero of the series, star each primed zero of the series, erase 
%         all primes and uncover every line in the matrix.  Return to Step 3.
%**************************************************************************

function [M,r_cov,c_cov,stepnum] = step5(M,Z_r,Z_c,r_cov,c_cov)

  zflag = 1;
  ii = 1;
  while zflag 
    % Find the index number of the starred zero in the column
    rindex = find(M(:,Z_c(ii))==1);
    if rindex > 0
      % Save the starred zero
      ii = ii+1;
      % Save the row of the starred zero
      Z_r(ii,1) = rindex;
      % The column of the starred zero is the same as the column of the 
      % primed zero
      Z_c(ii,1) = Z_c(ii-1);
    else
      zflag = 0;
    end
    
    % Continue if there is a starred zero in the column of the primed zero
    if zflag == 1;
      % Find the column of the primed zero in the last starred zeros row
      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
  
  % UNSTAR all the starred zeros in the path and STAR all primed zeros
  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
  
  % Clear the covers
  r_cov = r_cov.*0;
  c_cov = c_cov.*0;
  
  % Remove all the primes
  M(M==2) = 0;

stepnum = 3;

% *************************************************************************
% STEP 6: Add the minimum uncovered value to every element of each covered
%         row, and subtract it from every element of each uncovered column.  
%         Return to Step 4 without altering any stars, primes, or covered lines.
%**************************************************************************

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)

  % Step 2
    [r_cov,c_cov,M,stepnum] = step2(Edge);
  % Step 3
    [c_cov,stepnum] = step3(M,length(Edge));
  % Step 4
    [M,r_cov,c_cov,Z_r,Z_c,stepnum] = step4(Edge,r_cov,c_cov,M);
  % Calculate the deficiency
    cnum = length(Edge)-sum(r_cov)-sum(c_cov);