We study the problem of finding large weakly stable matchings when preference lists are incomplete and contain one-sided ties. Computing maximum weakly stable matchings is known to be NP-hard. We present a polynomial-time algorithm that achieves an improved approximation ratio of 1 + 1/e. Like a number of existing approximation algorithms for this problem, our algorithm is based on a proposal process in which numerical priorities are adjusted according to the solution of a linear program, and are used for tiebreaking purposes. Our main idea is to use an infinitesimally small step size for incrementing the priorities. Our analysis involves solving an infinite-dimensional factor-revealing linear program. We also show that the ratio 1 + 1/e is an upper bound for the integrality gap, which matches the known lower bound.

Consider a complete weighted bipartite graph G in which each left vertex u has two real numbers intercept and slope, each right vertex v has a real number quality, and the weight of any edge (u, v) is defined as the intercept of u plus the slope of u times the quality of v. Let m (resp., n) denote the number of left (resp., right) vertices, and assume that m geq n. We develop a fast algorithm for computing a maximum weight matching (MWM) of such a graph. Our algorithm begins by computing an MWM of the subgraph induced by the n right vertices and an arbitrary subset of n left vertices; this step is straightforward to perform in O(n log n) time. The remaining m - n left vertices are then inserted into the graph one at a time, in arbitrary order. As each left vertex is inserted, the MWM is updated. It is relatively straightforward to process each such insertion in O(n) time; our main technical contribution is to improve this time bound to O(sqrt{n} log^2 n). This result has an application related to unit-demand auctions. It is well known that the VCG mechanism yields a suitable solution (allocation and prices) for any unit-demand auction. The graph G may be viewed as encoding a special kind of unit-demand auction in which each left vertex u represents a unit-demand bid, each right vertex v represents an item, and the weight of an edge (u, v) represents the offer of bid u on item v. In this context, our fast insertion algorithm immediately provides an O(sqrt{n} log^2 n)-time algorithm for updating a VCG allocation when a new bid is received. We show how to generalize the insertion algorithm to update (an efficient representation of) the VCG prices within the same time bound.