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Functions for the rectangular assignment problem

Functions for the rectangular assignment problem

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14 Dec 2004 (Updated )

This package provides m- and mex-functions for solving the rectangular assignment problem.

assignmentallpossible(distMatrix)
function [assignment, cost] = assignmentallpossible(distMatrix)
%ASSIGNMENTALLPOSSIBLE    Compute solution of assignment problem
%   ASSIGNMENTALLPOSSIBLE(DISTMATRIX) computes the optimal assignment
%   (minimum overall costs) for the given rectangular distance or cost
%   matrix, for example the assignment of tracks (in rows) to observations
%   (in columns). The result is a column vector containing the assigned
%   column number in each row (or 0 if no assignment could be done).
%
%   [ASSIGNMENTALLPOSSIBLE, COST] = ASSIGNMENTALLPOSSIBLE(DISTMATRIX)
%   returns the assignment vector and the overall cost.
%
%   The distance matrix may contain infinite values (forbidden
%   assignments). Internally, the infinite values are set to a very large
%   finite number, so that the algorithm itself works on finite-number
%   matrices. Before returning the assignment, all assignments with
%   infinite distance are deleted (i.e. set to zero).
%
%   The algorithm recursively steps over all possible assignment paths. It
%   is very slow especially for large matrix dimensions. With this, it is
%   only suited for computing a reference solution.
%
%   <a href="assignment.html">assignment.html</a>  <a href="http://www.mathworks.com/matlabcentral/fileexchange/6543">File Exchange</a>  <a href="https://www.paypal.com/cgi-bin/webscr?cmd=_s-xclick&hosted_button_id=EVW2A4G2HBVAU">Donate via PayPal</a>
%
%   Markus Buehren
%   Last modified 05.07.2011

[nOfRows, nOfColumns] = size(distMatrix);

% check for infinite values
finiteIndex = isfinite(distMatrix);
if isempty(find(~finiteIndex, 1))
  % no infinite values contained in distMatrix
  % initialize maxCost with suboptimal algorithm
  [suboptimalassignment, maxCost] = assignmentsuboptimal2(distMatrix);
  infValue = inf;
  
else
  % distMatrix contains infinite values
  originalDistMatrix = distMatrix;
  finiteIndex = isfinite(distMatrix);
  index = find(~finiteIndex);
  if ~isempty(index)
    maxFiniteValue = max(max(distMatrix(finiteIndex)));
    if maxFiniteValue > 0
      infValue = abs(10 * maxFiniteValue * nOfRows * nOfColumns);
    else
      infValue = 10;
    end
    if isempty(infValue)
      % all elements are infinite
      assignment = zeros(nOfRows, 1);
      cost       = 0;
      return
    end   	
    distMatrix(index) = infValue;
  end
  maxCost = inf;	
end

if nOfRows <= nOfColumns
  
  [assignment, cost] = checksubtree(distMatrix, 0, maxCost, nOfRows, nOfColumns);
  
  if isempty(assignment)
    % suboptimal solution was equal to optimal solution
    assignment = suboptimalassignment;
  end

else
  
  % use transposed matrix
  [assignment, cost] = checksubtree(distMatrix.', 0, maxCost, nOfColumns, nOfRows);
  
  if isempty(assignment)
    % suboptimal solution was equal to optimal solution
    assignment = suboptimalassignment;
  else
    % switch index
    assignment = switchindex(assignment, nOfRows, nOfColumns);
  end
  
end

if cost >= infValue
  % remove invalid assignments
  distMatrix  = originalDistMatrix;
  rowIndex    = find(assignment);
  costVector  = distMatrix(rowIndex + nOfRows * (assignment(rowIndex)-1));
  finiteIndex = isfinite(costVector);
  cost = sum(costVector(finiteIndex));
  assignment(rowIndex(~finiteIndex)) = 0;
end

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function newAssignment = switchindex(assignment, nOfRows, nOfColumns)

newAssignment             = zeros(nOfRows, 1);
newAssignment(assignment) = (1:nOfColumns);

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function [newAssignment, newCost] = checksubtree(distMatrix, curCost, maxCost, nOfRows, nOfColumns)

if nOfRows > 1
  
  newCost       = maxCost;
  newAssignment = [];
  for n=1:nOfColumns
    thisCost = distMatrix(1,n) + curCost;
    if thisCost <= newCost
      
      % recursively pass distMatrix except the current row and column
      colIndex = [1:n-1,n+1:nOfColumns]';
      [thisAssignment, thisCost] = checksubtree(distMatrix(2:nOfRows, colIndex), thisCost, newCost, nOfRows-1, nOfColumns-1);
      
      if (thisCost <= newCost) && ~isempty(thisAssignment)
        newAssignment = [n; colIndex(thisAssignment)];
        newCost       = thisCost;
      end
    end	
  end
  
else
  
  [minDist, newAssignment] = min(distMatrix(1,:));
  newCost = minDist + curCost;
  
  if newCost > maxCost
    newAssignment = [];
  end	
  
end

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