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function [a,ass] = bipartiteMatchingIntProg(dst, nmatches)
% BIPARTITEMATCHINGINTPROG Use binary integer programming (linear objective) to solve for optimal linear assignment
% function a = bipartiteMatchingIntProg(dst)
% a(i) = best matching column for row i
%
% This gives the same result as bipartiteMatchingHungarian.
%
% function a = bibpartiteMatchingIntProg(dst, nmatches)
% only matches the specified number (must be <= min(size(dst))).
% This can be used to allow outliers in both source and target.
%
% For details, see Marciel & Costeira, "A global solution to sparse correspondence
% problems", PAMI 25(2), 2003
if nargin < 2, nmatches = []; end
[p1 p2] = size(dst);
p1orig = p1; p2orig = p2;
dstorig = dst;
if isempty(nmatches) % no outliers allowed (modulo size difference)
% ensure matrix is square
m = max(dst(:));
if p1<p2
dst = [dst; m*ones(p2-p1, p2)];
elseif p1>p2
dst = [dst m*ones(p1, p1-p2)];
end
end
[p1 p2] = size(dst);
c = dst(:); % vectorize cost matrix
% row-sum: ensure each column sums to 1
A2 = kron(eye(p2), ones(1,p1));
b2 = ones(p2,1);
% col-sum: ensure each row sums to 1
A3 = kron(ones(1,p2), eye(p1));
b3 = ones(p1,1);
if isempty(nmatches)
% enforce doubly stochastic
A = [A2; A3];
b = [b2; b3];
Aineq = zeros(1, p1*p2);
bineq = 0;
else
nmatches = min([nmatches, p1, p2]);
Aineq = [A2; A3];
bineq = [b2; b3]; % row and col sums <= 1
A = ones(1,p1*p2);
b = nmatches; % total num matches = b (otherwise get degenerate soln)
end
ass = bintprog(c, Aineq, bineq, A, b);
ass = reshape(ass, p1, p2);
a = zeros(1, p1orig);
for i=1:p1orig
ndx = find(ass(i,:)==1);
if ~isempty(ndx) & (ndx <= p2orig)
a(i) = ndx;
end
end
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