arXiv subjects

Yizhou Ai

Publications and source records attributed to Yizhou Ai.

4 recordsLinked to original sources

arXiv2026

The Complexity of Membership, Uniqueness, and Counting for Optimal Proportional Approval Voting Committees

Proportional Approval Voting (PAV) chooses committees that maximize a sum of harmonic utilities. We study the set of maximizing committees: whether a candidate belongs to some or all of them, whether the optimum is unique, and how many optima exist. When the committee size is part of the input, the three decision problems are $\Delta_2^\mathrm{P}$-complete. Uniqueness remains hard for instances with at most two optimal committees. Counting optimal committees is $\#\!\cdot\!\mathrm{OptP}$-complete under metric reductions: every function $f$ in this class reduces to an election with exactly $f(x)+1$ optimal committees. The reductions encode satisfying assignments directly as committees and use harmonic marginal rewards to realize binary objectives with polynomially many voters. Each satisfying assignment has a unique committee representation, and fixed clause ballots give these representations the same clause score. We also prove Turing equivalence with $\#\mathrm{SAT}$ and show that membership of the counting problem in $\#\mathrm{P}$ would imply $\\mathrm{NP}=\mathrm{coNP}$.

cs.GT↗

arXiv2026

Multi-Winner Elections: Justified Representation, Strategyproofness, and Risk-Avoiding Truthfulness

We study approval-based multi-winner elections with justified representation (JR) when voters strategically report their ballots. We prove that there does not exist a strategy-proof mechanism that outputs JR committees, even when the mechanism can be randomized and only ex-ante strategy-proofness is required. The impossibility result holds for any fixed voter's monotone utility function. In addition, our impossibility result continues to hold for even more restrictive settings, such as the setting where we are allowed to select fewer than $k$ candidates, with only 4 candidates and 3 voters. Motivated by our negative results, we then ask for weaker strategy-proof guarantees under voters' partial information. We use the notion of RAT-degree proposed by Hartman, Segal-Halevi, and Tao (EC'25), the number of other voters whose information a manipulator must know before a safe and profitable deviation is possible. Standard proportional rules, such as greedy approval voting, proportional approval voting (PAV), and the method of equal shares (MES), have poor performances under the RAT-degree metric (with low RAT-degrees).

cs.GT↗

arXiv2026

Full Justified Representation under Hare and Droop Quotas in Polynomial Time

I study Full Justified Representation (FJR) in approval-based multiwinner elections under both the Hare and Droop quota conventions. I introduce a descending-budget algorithm in which voters distribute their remaining budgets across their current representation gaps and candidates are purchased whenever the resulting offers cover a common price. With candidate price $\lambda_H=n/k$, the algorithm returns a Hare-FJR committee; with candidate price $\lambda_D=n/(k+1)$, it returns a committee satisfying the more demanding Droop-FJR axiom of Casey and Elkind. The two guarantees share a historical-payment invariant and a terminal row--column accounting argument, while the Droop proof requires a new residual-budget argument when all $k$ paid seats are filled. Both variants are deterministic once the voter and candidate orders are fixed and use $O(kmn)$ rational operations.

cs.GT↗

arXiv2026

Computational Complexity of Strong and Average Justified Representation

We study the approval-based multiwinner election problem where a set of $n$ voters cast approval-based ballots to a set of $m$ candidates, and we are to select a winner committee consisting of $k$ candidates. We consider two axioms: strong justified representation (SJR) and average justified representation (AJR). A winner committee satisfies SJR if the satisfaction for each voter in every $\ell$-cohesive group is at least $\ell$. AJR is a weaker axiom that requires the average satisfaction for each $\ell$-cohesive group to be at least $\ell$. It is well known that a winner committee satisfying AJR may not exist (and neither does SJR). In this paper, we study the computational complexity of the following decision problem: given an approval-based multiwinner election instance, decide if there exists a winner committee satisfying SJR/AJR. We prove that this problem is $\Theta_2^p$-complete for SJR, and $\Sigma_2^p$-complete for AJR. Our results indicate that the decision problem with SJR is more amenable to SAT-based implementations, whereas the decision problem with AJR is substantially harder. As byproducts, we derive some results that are interesting in their own right. Firstly, we show that adding one more adaptive query to an NP oracle on top of polynomially many non-adaptive NP queries does not add more computational power, and the resulting complexity class is still $\Theta_2^p$. Secondly, we construct a set system that can be useful in other applications, especially when doing reductions from typical satisfiability problems such as 3SAT.

cs.GT↗