Suppose that each member of a set A of applicants ranks a subset of a set P of posts in an order of preference, possibly involving ties. A matching is a set of (applicant, post) pairs such that each applicant and each post appears in at most one pair. A rank-maximal matching is one in which the maximum possible number of applicants are matched to their first choice post, and subject to that condition, the maximum possible number are matched to their second choice post, and so on. This is a relevant concept in any practical matching situation and it was first studied by Irving [2003].We give an algorithm to compute a rank-maximal matching with running time O (min( n + C , C √ n ) m ), where C is the maximal rank of an edge used in a rank-maximal matching, n is the number of applicants and posts and m is the total size of the preference lists.

Abstract. In this paper we consider the problem of computing a minimum cycle basis in a graph G with m edges and n vertices. The edges of G have non-negative weights on them. The previous best result for this problem was an O(m ω n) algorithm, where ω is the best exponent of matrix multiplication. It is presently known that ω < 2.376. We obtain an O(m 2 n + mn 2 log n) algorithm for this problem. Our algorithm also uses fast matrix multiplication. When the edge weights are integers, we have an O(m 2 n) algorithm. For unweighted graphs which are reasonably dense, our algorithm runs in O(m ω ) time. For any > 0, we also design a 1 + approximation algorithm to compute a cycle basis which is at most 1 + times the weight of a minimum cycle basis. The running time of this algorithm is O( m ω log(W/ )) for reasonably dense graphs, where W is the largest edge weight.

An instance of the stable marriage problem is an undirected bipartite graph G = (X∪ W, E) with linearly ordered adjacency lists with ties allowed in the ordering. A matching M is a set of edges no two of which share an endpoint. An edge e = (a, b) ∈ E \ M is a blocking edge for M if a is either unmatched or strictly prefers b to its partner in M , and b either is unmatched or strictly prefers a to its partner in M or is indifferent between them. A matching is strongly stable if there is no blocking edge with respect to it. We give an O(nm) algorithm for computing strongly stable matchings, where n is the number of vertices and m is the number of edges. The previous best algorithm had running time O(m 2 ). We also study this problem in the hospitals-residents setting, which is a many-to-one extension of the above problem. We give an O(m P h∈H p h ) algorithm for computing a strongly stable matching in the hospitals-residents problem, where p h is the quota of a hospital h. The previous best algorithm had running time O(m 2 ).