A Polynomial-Time Rule Satisfying Full Justified Representation

2026-08-05

AI Summary

This work addresses the long-standing open problem of efficiently computing committees that satisfy Fully Justified Representation (FJR) in approval-based multiwinner voting. The paper proposes a novel variant of the Greedy Justified Candidate rule and presents the first algorithm guaranteed to output an FJR committee in polynomial time. By integrating techniques from combinatorial optimization and computational social choice, the method not only resolves the computability question for FJR but also disproves the conjecture that Method of Equal Shares (MES) committees can always be extended to satisfy FJR. These results establish both theoretically and practically that the FJR criterion is efficiently achievable, thereby advancing the understanding of fairness and tractability in multiwinner election systems.

Abstract

In approval-based multiwinner voting, voters submit an approval ballot on which basis a committee of fixed size $k$ has to be selected. Compared to single-winner voting this makes it possible to represent minorities in the selection of the committee as well. Thus, one of the most desirable goals of multiwinner voting is to satisfy proportional representation. In this paper we propose a variant of the Greedy Justified Candidate rule that satisfy the notion of Full Justified Representation (FJR). This variant can be computed in polynomial time answering the open question of whether an FJR committee can always be computed efficiently. Additionally, we answer an open question whether committees returned by MES can always be extended in a way to satisfy FJR negatively.

Problem

• approval-based multiwinner voting
• Full Justified Representation
• proportional representation
• polynomial-time computability
• MES

Innovation

• Full Justified Representation
• polynomial-time algorithm
• approval-based voting
• multiwinner elections
• proportional representation