How to design fair and (computationally) efficient voting rules is a central challenge in Computational Social Choice. In this paper, we aim at designing efficient algorithms for computing most equitable rules for large classes of preferences and decisions, which optimally satisfy two fundamental fairness/equity axioms: anonymity (every voter being treated equally) and neutrality (every alternative being treated equally). By revealing a natural connection to the graph isomorphism problem and leveraging recent breakthroughs by Babai [2019], we design quasipolynomial-time algorithms that compute most equitable rules with verifications, which also compute verifications about whether anonymity and neutrality are satisfied at the input profile. Further extending this approach, we propose the canonical-labeling tie-breaking, which runs in quasipolynomial-time and optimally breaks ties to preserve anonymity and neutrality. As for the complexity lower bound, we prove that even computing verifications for most equitable rules is GI-complete (i.e., as hard as the graph isomorphism problem), and sometimes GA-complete (i.e., as hard as the graph automorphism problem), for many commonly studied combinations of preferences and decisions. To the best of our knowledge, these are the first problems in computational social choice that are known to be complete in the class GI or GA.

翻译：如何设计公平且（计算上）高效的投票规则是计算社会选择领域的核心挑战。本文旨在为广泛的偏好与决策类别设计高效算法，以计算最优满足两大基本公平/公正公理——匿名性（每位选民被平等对待）与中立性（每个备选方案被平等对待）——的最公平规则。通过揭示其与图同构问题的内在关联，并借助Babai [2019]的最新突破性成果，我们设计了拟多项式时间算法来计算带验证的最公平规则，该算法同时能验证输入偏好剖面是否满足匿名性与中立性。进一步扩展此方法，我们提出了规范标号平局决胜机制，该机制在拟多项式时间内运行，并能以最优方式打破平局以保持匿名性与中立性。在复杂度下界方面，我们证明对于许多常见研究的偏好与决策组合，即使仅计算最公平规则的验证问题也是GI完全的（即与图同构问题同等困难），有时甚至是GA完全的（即与图自同构问题同等困难）。据我们所知，这是计算社会选择领域中首个被证明属于GI类或GA类完全问题的问题。