The paper presents several results on the maximum stable matching problem with matroids and partial orders:
It introduces a new theorem on basis exchanges for two disjoint bases of a matroid, which is used as a key tool to generalize the 1.5-approximation algorithm for the maximum stable marriage problem with ties to the matroid kernel problem.
It shows that there exists a simple 1.5-approximation algorithm for the maximum matroid kernel problem when the preferences are given as interval orders, a broad subclass of partial orders. This extends the previous 1.5-approximation results for the maximum stable marriage problem with ties.
It proves that for general partial orders, it is NP-hard to approximate the maximum stable marriage problem within a factor better than 2 assuming the Unique Games Conjecture. This complements the previously known hardness results for the maximum stable marriage problem with ties.
It shows that the integrality gap of the natural LP relaxation of the maximum stable marriage problem with general partial orders is at least 2, while it is at most 1.5 for the case of interval orders.
The results suggest that the class of interval orders may be the right generalization for which a nontrivial approximation is still possible for the maximum stable matching problem.
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