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Strong core and Pareto-optimal solutions for the multiple partners matching problem under lexicographic preferences

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  • P'eter Bir'o
  • Gergely Cs'aji

Abstract

In a multiple partners matching problem the agents can have multiple partners up to their capacities. In this paper we consider both the two-sided many-to-many stable matching problem and the one-sided stable fixtures problem under lexicographic preferences. We study strong core and Pareto-optimal solutions for this setting from a computational point of view. First we provide an example to show that the strong core can be empty even under these severe restrictions for many-to-many problems, and that deciding the non-emptiness of the strong core is NP-hard. We also show that for a given matching checking Pareto-optimality and the strong core properties are co-NP-complete problems for the many-to-many problem, and deciding the existence of a complete Pareto-optimal matching is also NP-hard for the fixtures problem. On the positive side, we give efficient algorithms for finding a near feasible strong core solution, where the capacities are only violated by at most one unit for each agent, and also for finding a half-matching in the strong core of fractional matchings. These polynomial time algorithms are based on the Top Trading Cycle algorithm. Finally, we also show that finding a maximum size matching that is Pareto-optimal can be done efficiently for many-to-many problems, which is in contrast with the hardness result for the fixtures problem.

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  • P'eter Bir'o & Gergely Cs'aji, 2022. "Strong core and Pareto-optimal solutions for the multiple partners matching problem under lexicographic preferences," Papers 2202.05484, arXiv.org.
    • Handle: RePEc:arx:papers:2202.05484
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    References listed on IDEAS

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