发表机构
Indian Institute of Technology Kharagpur; Indian Institute of Science; University of Sheffield（印度理工学院卡拉格普尔分校; 印度科学学院; 谢菲尔德大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对基于选区的选举，提出一种自适应随机算法，以约 O~(1/eps^2 log m/del log 1/del) 次查询预测获胜者，改进现有结果并接近理论下界。
AI 中文摘要
在基于选区的选举中，N 名选民被划分为 k 个选区，每名选民投票给 m 名候选人中的一名。每个选区使用多数制规则（即获得最多票数的候选人被宣布为获胜者，平局时按某种固定规则打破）选出一名获胜者，而总体获胜者通过对选区获胜者应用多数制来确定；我们假设在选区获胜者中存在唯一的获胜者。此类选举的胜利幅度是必须改变的最少票数，以使当前获胜者不再是唯一的选区获胜者。我们研究在查询复杂度模型中预测基于选区的选举获胜者的问题，在该模型中，可以查询单个选民的投票。目标是尽量减少查询次数。这种设置涵盖了出口民调，其中查询对应于采访选民，并且与查询复杂度和性质测试中的问题密切相关。假设选举的胜利幅度至少为 eps N，Dey、Kar 和 Sanyal（AAMAS 2023）针对两名候选人的情况给出了错误概率为 del、查询复杂度为 tilde{O}(1/eps^6 log^2 1/del) 的算法，在额外假设选区人口均衡的情况下，该复杂度改进为 tilde{O}(1/eps^4 log^2 1/del)。我们的主要结果是一种自适应随机算法，对于任意基于选区的选举和任何错误参数 del，以至少 1-del 的概率，使用 tilde{O}(1/eps^2 log m/del log 1/del) 次查询正确预测获胜者。特别是，我们改进了 Dey 等人针对任意选区人口的界限，并将其结果扩展到任意数量的候选人。此外，对于常数数量的候选人，我们的算法几乎匹配 Omega(1/eps^2 log 1/del) 的查询复杂度下界，该下界即使对于两名候选人和单个选区也成立。
英文摘要
In a district-based election, N voters are partitioned into k districts, and each voter votes for one of m candidates. Each district elects a winner using the plurality rule (i.e. the candidate getting the largest number of votes is declared the winner, breaking ties as per some fixed rule), and the overall winner is determined by applying plurality to the district winners; we assume that there is a unique winner amongst the district winners. The margin of victory of such an election is the minimum number of votes that must be altered so that the current winner ceases to be the unique district winner. We study the problem of predicting the winner of a district-based election in the query complexity model, where one has query access to individual votes. The objective is to minimise the number of queries. This setting captures exit polling, where queries correspond to interviewing voters, and is closely related to problems in query complexity and property testing. Assuming that the margin of victory of the election is at least eps N, Dey, Kar and Sanyal (AAMAS 2023) gave algorithms for the case of two candidates with error probability del and query complexity tilde{O}(1/eps^6 log^2 1/del), which improves to tilde{O}(1/eps^4 log^2 1/del) under the additional assumption that district populations are balanced. Our main result is an adaptive randomised algorithm that, for an arbitrary district-based election and any error parameter del, with probability at least 1-del, predicts the winner correctly using tilde{O}(1/eps^2 log m/del log 1/del) queries. In particular, we improve the bounds of Dey et al. for arbitrary district populations and extend their results to any number of candidates. Furthermore, for constantly many candidates, our algorithm nearly matches a lower bound of Omega(1/eps^2 log 1/del) on the query complexity that holds even for two candidates and a single district.
