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.

We give a $\frac{3}{2}$ -approximation algorithm for finding stable matchings that runs in O(m) time. The previous most well-known algorithm, by McDermid, has the same approximation ratio but runs in O(n^3/2m) time, where n denotes the number of people andm is the total length of the preference lists in a given instance. In addition, the algorithm and the analysis are much simpler. We also give the extension of the algorithm for computing stable many-to-many matchings.

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 is unmatched, 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 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 aforementioned problem. We give an O ( m ∑ _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 ).

We consider the problem of computing a minimum cycle basis of an undirected non-negative edge-weighted graph G with m edges and n vertices. In this problem, a {0,1} incidence vector is associated with each cycle and the vector space over \(\mathbb{F}_{2}\) generated by these vectors is the cycle space of G. A set of cycles is called a cycle basis of G if it forms a basis for its cycle space. A cycle basis where the sum of the weights of the cycles is minimum is called a minimum cycle basis of G. Minimum cycle basis are useful in a number of contexts, e.g. the analysis of electrical networks and structural engineering. The previous best algorithm for computing a minimum cycle basis has running time O(m ^ω n), where ω is the best exponent of matrix multiplication. It is presently known that ω0, we also design an 1+ε approximation algorithm. The running time of this algorithm is O((m ^ω /ε)log (W/ε)) for reasonably dense graphs, where W is the largest edge weight.