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.
Lam and Plaxton REF studied the problem of finding large weakly stable matchings when preference lists are incomplete and contain onesided ties and presented a 1 + 1/e-factor approximation algorithm.
A (1 + 1/e)-Approximation Algorithm for Maximum Stable Matching with One-Sided Ties and Incomplete Lists
SODA
Computer Science