Polynomial-time computation of an FJR-satisfying outcome for approval-based committee elections

Determine whether there exists a polynomial-time computable voting rule that always produces an outcome satisfying Fully Justified Representation for committee elections with approval preferences and unit costs.

Background

Fully Justified Representation (FJR) is the strongest of the three justified-representation axioms studied in the paper and implies both Extended Justified Representation (EJR) and Proportional Justified Representation (PJR). Although FJR-satisfying outcomes are known to exist and can be generated by the Greedy Cohesive Rule, computing such an outcome is generally difficult. The paper identifies the existence of a polynomial-time rule for the special case of approval preferences and equal costs as a longstanding unresolved problem.

References

FJR is stronger than EJR even for committee elections with approval preferences; whether a polynomial time computable voting rule satisfying FJR exists is an important open problem.

— The Complexity of Justified Representation with Additive Utilities  (2609.02030 - Baharav et al., 2 Sep 2026) in Introduction, paragraph beginning “Moreover, Peters et al. introduce…”

We believe it is an interesting open question whether a sequential rule can satisfy FJR for approval utilities and unit costs.

— The Complexity of Justified Representation with Additive Utilities  (2609.02030 - Baharav et al., 2 Sep 2026) in Section 7, Discussion, paragraph beginning “The result that finding an outcome satisfying FJR…”

It is an interesting open question whether a different polynomial-time, local search algorithm can overcome the barriers for sequential algorithms.

— The Complexity of Justified Representation with Additive Utilities  (2609.02030 - Baharav et al., 2 Sep 2026) in Section 7, Discussion, paragraph beginning “While not satisfying a justified representation axiom…”