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We study the problem of finding large weakly stable matchings when preference lists are incomplete and contain onesided 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.

A (1 + 1/e)-Approximation Algorithm for Maximum Stable Matching with One-Sided Ties and Incomplete Lists