Dynamic matchings and quasidynamic fractional matchings. II

Consider a directed graph G in which every edge has an associated real-valued distance and a real-valued weight. The weight of an undirected circuit of G is the sum of the weights of the edges, whereas the distance of an undirected circuit is the sum of the distances of the forward edges of the circuit minus the sum of the distances of the backward edges. A trivial circuit is a two-edge circuit in which one edge of G appears twice on the circuit. A quasidynamic fractional matching (or Q-matching) is a collection of vertex-disjoint circuits such that each circuit is either trivial or else it is an odd circuit whose distance is nonzero. The Q-matching problem is to find a Q-matching that maximizes the sum of the weights of its circuits. The Q-matching problem generalizes both the matching problem and the fractional matching problem. Moreover, the dynamic matching problem, which is a matching problem on an infinite dynamic (time-expanded) graph, is linearly transformable to the Q-matching problem, as shown in Part I of this paper. In this paper we solve the Q-matching problem by generalizing Edmonds' blossom algorithm. In fact, all of the major components of the blossom algorithm-including alternating trees, augmentations, shrinking, and expanding-are appropriately generalized to yield a running time that is proportional to that for the weighted matching problem. Furthermore, if all edge distances are equal to zero, this new algorithm reduces to the blossom algorithm.