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Finding All Stable Pairs and Solutions to the Many-to-Many Stable Matching Problem

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Abstract

The many-to-many stable matching problem (MM), defined in the context of a job market, asks for an assignment of workers to firms satisfying the quota of each agent and being stable, pairwise or setwise, with respect to given preference lists or relations. In this paper, we propose a time-optimal algorithm that identifies all stable worker--firm pairs and all stable assignments under pairwise stability, individual preferences, and the max-min criterion. We revisit the poset graph of rotations to obtain an optimal algorithm for enumerating all solutions to the MM and an improved algorithm finding the minimum-weight one. Furthermore, we establish the applicability of all aforementioned algorithms under more complex preference and stability criteria. In a constraint programming context, we introduce a constraint that models the MM and an encoding of the MM as a constraint satisfaction problem. Finally, we provide a series of computational results, including the case where side constraints are imposed.

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... There is a broad literature on algorithms for MM. In brief, Baïou and Balinski [6] provide an algorithm that finds an MM solution and identifies some non-stable pairs (i.e., pairs of agents appearing in no solution), while Eirinakis et al. [12] provide an algorithm for finding all such pairs. Bansal et al. [10] present algorithms for the egalitarian as well as the 'optimal' stable matching, while faster such algorithms are proposed in [12]. ...
... In brief, Baïou and Balinski [6] provide an algorithm that finds an MM solution and identifies some non-stable pairs (i.e., pairs of agents appearing in no solution), while Eirinakis et al. [12] provide an algorithm for finding all such pairs. Bansal et al. [10] present algorithms for the egalitarian as well as the 'optimal' stable matching, while faster such algorithms are proposed in [12]. ...
... In such cases, our result provides a minimal linear relaxation, which admits additional constraints per variant, and can be used efficiently in the framework of general solution methods (e.g., Branch & Cut). In contrast, the problemspecific combinatorial algorithms that also minimize a linear function over MM [12] become non-applicable in the presence of additional constraints. Hence, we consider a minimal linear description of stable b-matchings valuable in terms of solving several related optimization problems. ...
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The realization of stable b-matchings as matroid kernels yields the existing linear description of the stable b-matching (MM) problem. We revisit that description to derive the dimension, the facets, and the minimum equation system of the stable b-matching polytope. The derived minimal description includes O(m) constraints, m being the number of pairs, thus being significantly smaller than the existing one and linear with respect to the problem size. The result carries over to the stable admissions (SA) problem, whose existing linear description relies on an exponential number of comb inequalities; i.e., we identify O(m) among these inequalities that define a minimal description of SA.
... In 2012 Eirinakis et al. [15] used the poset graph of rotations to enumerate all solutions of HR, and presented an improved version of the direct CP model of Manlove et al. [35]. Subsequently, Siala and O'Sullivan [47] used the rotation poset to model stable matchings as SAT formulation for all three types of problems: one-to-one, one-to-many, and many-tomany. ...
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... 6 Among other results, they use cycles to compute the full set of stable match-ings. Eirinakis et al. (2012) revise and improve the algorithm presented in Bansal et al. (2007). Moreover, they extend the algorithm for a model in which agents' preferences satisfy the "max-min criteria". ...
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Some Examples The Logic of Propositions The Logic of Discrete Variables The Logic of 0-1 Inequalities Cardinality Clauses Classical Boolean Methods Logic-Based Modeling Logic-Based Branch and Bound Constraint Generation Domain Reduction Constraint Programming Continuous Relaxations Decomposition Methods Branching Rules Relaxation Duality Inference Duality Search Strategies Logic-Based Benders Decomposition Nonserial Dynamic Programming Discrete Relaxations References Index.
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We propose a general definition of stability, setwise-stability, and show that it is a stronger requirement than pairwise-stability and core. We also show that the core and the set of pairwise-stable matchings may be non-empty and disjoint and thus setwise-stable matchings may not exist. For many labor markets the effects of competition can be characterized by requiring only pairwise-stability. For such markets we define substitutability and we prove the existence of pairwise-stable matchings. The restriction of our proof to the College Admission Model is simple and short and provides an alternative proof for the existence of stable matchings for this model.
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ILOG S.A., ILOG Solver Callable Library 6.1, 2005.