On the Stability of Multiple Partner Stable Marriages with Ties

We consider the generalized version of the stable marriage problem where each man and woman’s preference list may have ties. Furthermore, each man and woman wishes to be matched to as many of acceptable partners as possible, up to his or her specified quota. Many-to-many version of the stable marriage problem has wide applications in matching retailers and shopkeepers in e-marketplaces. We investigate different forms of stability in this context and describe an algorithm to find strongly stable matchings (if one exists) in the context of multiple partner stable marriage problem with ties. In the context of Hospital-Residents problem for which only the resident-oriented algorithm for finding a strongly stable matching is known, this algorithm gives a hospital-oriented version (for the same) as well. Furthermore, in any instance of many-to-many stable marriage problem with ties, we show that the set of strongly stable matchings forms a distributive lattice. The results in this paper extend those already known for the one-to-one version and many-to-one version (Hospitals-Residents problem) of the problem.

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