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

TL;DR

This paper investigates the computational complexity of strong core and Pareto-optimal solutions in multi-partner matching problems with lexicographic preferences, providing both hardness results and efficient algorithms for approximate solutions.

Contribution

It establishes NP-hardness and co-NP-completeness results for core and Pareto-optimality, and introduces polynomial algorithms for near-feasible solutions and maximum Pareto-optimal matchings.

Findings

Strong core can be empty even under severe restrictions.

Deciding non-emptiness of the strong core is NP-hard.

Efficient algorithms exist for near-feasible strong core solutions and maximum Pareto-optimal matchings.

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.