Just read this blog to know such a puzzler has been going on and to understand why John came to me to ask for the use of the munkres code a few weeks ago. Thanks to Darren who pointed out my code as a solution to the puzzler. Thank you and John to post this puzzler, which shows how useful of the Hungarian method is.

Regards,
Yi

For some reason, Francois’ comment needed to be “moderated” because the automatic spam filter thought it might be spammy. Usually things are posted immediately, but sometimes things get stuck, like his post.

Doug

I like how the community of readers are able to quickly write, optimize and re-optimize a problem. Petter makes a solid greedy algorithm, then Danilo is able to optimize the bottleneck in it.

Francois is able to make an algorithmic advance by recognizing the problem as one with an existing solution and then Darren points out a file exchange submission that does this already.

This is what I love about the social computing aspect of this site.

Thanks for all the effort!
-Doug

The greedy algorithm trick is to use sort. If we have

[junk,tags] = sort(A,2);

then if the first column of tags contains all of the numbers from 1:n, then the solution is given by that column. This is extremely fast to compute, and so is well worth checking. Even if not, the sort does much of the work that jensen was doing. You only need to look for the columns that were not assigned positions uniquely by the sort.

Regardless, munkres by Yi Cao is an order of magnitude better than my own search scheme. My thanks to those who pointed it out.

 function [matOut, trace] = findTrace(matIn)
% Specify the problem as a linear program
N = size(matIn,1);
% Create the equality matrix
i = reshape(repmat(1:2*N, [N 1]), [], 1);
j = bsxfun(@plus, (1:N)', 0:N:N*N-1);
j = [reshape(j, [], 1); reshape(j', [], 1)];
Aeq = sparse(i, j, ones(size(i)), 2*N, N*N);
% Optimize
options = optimset('LargeScale', 'on', 'Display', 'off');
x = linprog(matIn(:), [], [], Aeq, ones(1, 2*N), zeros(N*N, 1), ones(N*N, 1), [], options);
x = reshape(x, N, N);
% Extract the permutation
[perm dummy] = find(x' > 0.5);
matOut = matIn(:,perm);
trace = diag(matOut);
% Check the solution is valid
assert(numel(perm)==N && numel(unique(perm(:)))==N);

 
function [matOut tr] = findTrace3(matIn)
[nr nc]    = size(matIn);
matOut     = matIn;
perm = zeros(nc, 1);
for n = 1 : nc
    [value cIndex]    	= min(min(matIn));
    rowNr            	= find(matIn(:,cIndex) == value, 1);
    perm(rowNr)	        = cIndex;
    matIn(rowNr,:)     = NaN;
end
matOut = matOut(:,perm);
tr = diag(matOut);
% Check the solution is valid
assert(numel(perm)==nc && numel(unique(perm(:)))==nc);