Weighted popular common independent sets with ties or partial orders

Determine whether polynomial-time algorithms for weighted popular matchings with ties and unweighted popular common independent sets with arbitrary partial-order preferences can be extended to the weighted popular common independent set problem under ties or arbitrary partial-order preferences.

Background

The paper develops a polynomial-time algorithm for weighted popular common independent sets when every applicant has a strict preference order. The authors note that broader preference structures, including ties and arbitrary partial orders, have already been treated for related models: weighted popular matching with ties and unweighted popular common independent sets with arbitrary partial-order preferences.

The unresolved question is whether these two extensions can be combined with the weighted popular common independent set framework. Such a result would generalize the paper's algorithm beyond strict preferences and unify the existing weighted and partial-order results.

References

For example, Mestre gave a polynomial-time algorithm for the weighted popular matching problem in the presence of ties, and Kavitha, Makino, Schlotter, and Yokoi gave a polynomial-time algorithm for the (unweighted) popular common independent set problem under arbitrary partial-order preferences. The question of whether these results can be extended to the weighted popular common independent set problem remains an interesting open problem.

— A Matroid Generalization of Weighted Popular Matching: Characterization and Algorithm  (2610.17744 - Iwamasa et al., 13 Oct 2026) in Section 6, Concluding Remarks