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 = ( w i j ) n × n of "affinity" data. Its goal is to find a permutation π over 1 : n whose total weight ? i w i , 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 u 1 , ? , u r , v 1 , ? v r ? R n such that W = ? i = 1 r u i v i T . In this paper, we partially address a question raised by David Karger who asked for a characterization of the structure of the maximum-weighted bipartite matchings when the rank of the affinity data is low. In particular, we study the following locality property: For an integer k 0 , we say that the bipartite matchings of G have locality at most k if for every sub-optimal matching π of G, there exists a matching ? of larger total weight that can be reached from π by an augmenting cycle of length at most k.We prove the following main theorem: For every W ? 0 , 1 n × n of rank r and ? ? 0 , 1 , there exists W ? ? 0 , 1 n × n such that (i) W ? has rank at most r + 1 , (ii) the entry-wise ∞-norm ? W - W ? ? ∞ ? ? , and (iii) the weighted bipartite graph with affinity data W ? has locality at most ? r / ? ? r . In contrast, this property is not true if perturbations are not allowed. We also give a tight bound for the binary case.