Maximum bipartite matchings with low rank data: Locality and perturbation analysis.

The maximum-weighted bipartite matching problem between two sets U=V=[1:n] is defined by a matrix W=(wij)n×n of “affinity” data. Its goal is to find a permutation π over [1:n] whose total weight ∑iwi,π(i) is maximized. In various practical applications,3 the affinity data may be of low rank (or approximately low rank): we say W has rank at most r if there are 2r vectors u1,…,ur,v1,…vr∈Rn such thatW=∑i=1ruiviT.