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In a multiple partners matching problem the agents can have multiple partnersup to their capacities. In this paper we consider both the two-sidedmany-to-many stable matching problem and the one-sided stable fixtures problemunder lexicographic preferences. We study strong core and Pareto-optimalsolutions for this setting from a computational point of view. First we providean example to show that the strong core can be empty even under these severerestrictions for many-to-many problems, and that deciding the non-emptiness ofthe strong core is NP-hard. We also show that for a given matching checkingPareto-optimality and the strong core properties are co-NP-complete problemsfor the many-to-many problem, and deciding the existence of a completePareto-optimal matching is also NP-hard for the fixtures problem. On thepositive side, we give efficient algorithms for finding a near feasible strongcore solution, where the capacities are only violated by at most one unit foreach agent, and also for finding a half-matching in the strong core offractional matchings. These polynomial time algorithms are based on the TopTrading Cycle algorithm. Finally, we also show that finding a maximum sizematching that is Pareto-optimal can be done efficiently for many-to-manyproblems, which is in contrast with the hardness result for the fixturesproblem.

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