New algorithm solves election fairness in polynomial time

Computer scientist Yizhou Ai presents a new deterministic algorithm for approval-based multiwinner elections that achieves Full Justified Representation (FJR) under both Hare and Droop quota systems in polynomial time. The descending-budget method has voters spread their remaining budgets across representation gaps, with candidates purchased once offers cover a common price. Using a Hare quota price returns a Hare-FJR committee, while a Droop quota price satisfies the more demanding Droop-FJR axiom. Both versions run in O(kmn) rational operations once voter and candidate orders are fixed, sharing a historical-payment invariant and a terminal row–column argument — with the Droop case adding a residual-budget step when all seats are filled.