  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.
