Reverse Sequential Proportional Approval Voting Rule: Proportionality and Approximation Guarantees

We study the computational complexity of winner determination problems in approval-based committee elections under Thiele voting rules. These form a class of rules parameterized by a fixed weight vector that specifies how a voter's satisfaction depends on the number of approved candidates elected. We first analyze the structure of optimal solutions based on the sets of voters who approve each candidate---that is, how voters' approval ballots induce dependencies between candidates---revealing constraints on a winning committee under any fixed Thiele voting rule. Using this, we design FPT algorithms for Proportional Approval Voting (PAV) and other Thiele rules on a natural restricted domain known as the Voter Interval (VI) domain---that is, after a suitable ordering of voters, each candidate is approved by a consecutive interval of voters. In particular, we show that every Thiele rule on VI is FPT with respect to a parameter for which the problem is NP-hard on general instances, even when the parameter takes constant values. Our results advance the understanding of the computational complexity of PAV on Voter Interval instances, which remains one of the central open questions in this area. We further resolve two open questions from the literature on PAV (and other Thiele voting rules) by providing a polynomial-time algorithm for instances where each candidate is approved by at most two voters, and an FPT algorithm parameterized by the total score of a winning committee.

Full justified representation (FJR) is among the strongest known satisfiable proportionality axioms for approval-based committee elections. Recent work has shown that an FJR committee can be found in polynomial time, but verifying whether a given committee satisfies FJR remains coNP-complete. We introduce FJR+, a strict strengthening of FJR and EJR+ that can be verified and satisfied in polynomial time. We then analyze the Residual-Budget Greedy (RBG) algorithm and prove that it selects a partial committee such that every size-$k$ completion satisfies FJR+. This freedom allows us to use sequential Phragmén to obtain a priceable completion. The resulting rule always satisfies FJR+ and the sub-core, and it is priceable whenever at least $k$ candidates receive an approval. We also obtain a Droop-quota version of FJR+. Finally, we extend FJR+ to approval-based participatory budgeting with arbitrary project costs. A project-specific version of RBG computes this property in polynomial time and can be continued to a priceable outcome satisfying a cost-based version of the sub-core.

We study proportional representation in the temporal voting model, where collective decisions are made repeatedly over time over a fixed horizon. Prior work has extensively investigated how proportional representation axioms from multiwinner voting (e.g., justified representation (JR) and its variants) can be adapted, satisfied, and verified in this setting. However, much less is understood about their interaction with social welfare. In this work, we quantify the efficiency cost of enforcing proportionality. We formalize the welfare-proportionality tension via the worst-case ratio between the maximum achievable utilitarian welfare and the maximum welfare attainable subject to a proportionality axiom. We show that imposing proportional representation in the temporal setting can incur a growing, yet sublinear, welfare loss as the number of voters or rounds increases. We further identify a clean separation among axioms: for JR, the welfare loss diminishes as the time horizon grows and vanishes asymptotically, whereas for stronger axioms this conflict persists even with many rounds. Moreover, we prove that welfare maximization under each axiom is NP-complete and APX-hard, even under static preferences and bounded-degree approvals, and provide fixed-parameter algorithms under several natural structural parameters.