The diameter of the stable matching (stable marriage) polytope is bounded from above by n 2 , where n is the number of men (or women) involved in the matching; this bound is attainable.
Introduction.The stable matching (SM) problem (also known as the stable marriage problem [10]) asks for a matching from a set M of men to a set W of women that is stable under given preference lists, i.e., when men have preferences with regard to women, and vice versa. In 1962, Gale and Shapley introduced the problem in the setting of college admissions [10] and opened a new research chapter in the fields of combinatorics, mathematical economics, and social behavior [15,12,20]. Its significance has been recognized in the faces of Roth and Shapley, who received the Nobel Prize in Economics "for the theory of stable allocations and the practice of market design" [17]. The current work adds another page to this chapter by studying the diameter of the polytope P S defined as the convex hull of incidence vectors of stable matchings.The polytope P S bears a wider interest from a combinatorial-theoretic point of view since, as shown in [4,13], every finite distributive lattice is a set of stable matchings. The polytope was initially investigated in [25], where a description in terms of linear inequalities is given. This description is simplified and extended in [21]. Further work in the same context includes [19,2,1,23,24,8]. In [18], the faces of P S are characterized in terms of the lattice of stable matchings with the use of a directed line graph, namely, the marriage graph Γ (see [18, Definition 6]). A minimal linear description of P S in terms of its minimal equation system as well as all the classes of facets is given in [7].The diameter of P S (diam(P S )) has been examined in neither of the abovementioned papers nor (to the best of our knowledge) in any other work appearing in the literature. We provide an upper bound on diam(P S ) equal to min{|M|,|W |}
Abstract. Consider a many-to-many matching market that involves two finite disjoint sets, a set of applicants A and a set of courses C. Each applicant has preferences on the different sets of courses she can attend, while each course has a quota of applicants that it can admit. In this paper, we examine Pareto optimal matchings (briefly POM) in the context of such markets, that can also incorporate additional constraints, e.g., each course bearing some cost and each applicant having an available budget. We provide necessary and sufficient conditions for a many-to-many matching to be Pareto optimal and show that checking whether a given matching is Pareto optimal requires O(|A| 2 · |C| 2 ) time. Moreover, we provide a generalized version of serial dictatorship, which can be used to obtain any many-to-many POM. We also study the problems of finding a minimum cardinality and a maximum cardinality POM. We show that the former is NP-complete even in one-to-one markets with the preference list of each applicant containing at most two entries. For the latter problem we show that, although it is polynomially solvable in the special one-to-one case, it is NP-complete for many-to-many markets.
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