Pareto Stable Mechanism and its Algorithm for Many to Many Matching Problem with Quota

This paper discusses the Pareto-stable mechanism and its algorithm for many to many matching problem with quota. Firstly, we propose a Pareto stable matching algorithm, this algorithm consists of two subroutines called modified TTC algorithm and eliminate PI cycle algorithm. Secondly, we propose a worker-optimal stable matching algorithm, it is operated in an iterative way of eliminating WPI cycles and WPI chains starting from an arbitrary Pareto-stable matching. Thirdly, these algorithms provide an alternative proof of the existence Pareto stable matchings and worker-optimal stable matchings for a many to many matching problem with quota under the minimal responsive preference restriction. Moreover, we propose these algorithms are bounded by O((nm)^3) time, where n and m are the numbers of workers and firms respectively, the algorithms complexity is lower than that of the state-of-the-art algorithms