发表机构
University of Edinburgh; Imperial College London; University of Oxford（爱丁堡大学; 伦敦帝国学院; 牛津大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对二元字符串覆盖问题及委托投票代表设计需求，研究了带缺失项的硬阈值二元$k$-中心问题的计算复杂度，将无缺失项的难度结果推广至任意中心数和大阈值场景。
AI 中文摘要
我们研究了用固定数量的中心字符串覆盖长度相同、可能存在缺失项的二元字符串的问题，要求每个输入字符串与最近中心的距离都在给定的相对距离范围内。该问题在二元$k$-中心聚类和生物信息学中均有应用，但我们的主要研究动机来自计算社会选择领域。在多议题赞成投票的场景下，我们考虑了一个先于任何选举的算法问题：应如何设计代表，才能让尽可能多的选民愿意委托投票？我们从中心数量和一致性阈值这两个因应用而异的参数维度研究该问题，并完整刻画了当存在缺失项时该问题的计算复杂度。对于无缺失项的特殊情况，我们将 hardness 结果推广到任意数量的中心和大阈值情形，仅留有小范围的阈值尚未解决。
英文摘要
We study the problem of covering binary strings of the same length, possibly with missing entries, by a fixed number of center strings, such that each input string is within a given relative distance of the closest center. The problem has applications in binary $k$-center clustering and bioinformatics, but our main motivation comes from computational social choice. In the setting of multi-issue approval voting, we consider an algorithmic question that precedes any election: how should representatives be designed so that as many voters as possible are willing to delegate? We study the problem in two dimensions, namely the number of centers and the agreement threshold, parameters that are application-specific, and provide a complete picture of its computational complexity when entries may be missing. For the special case without missing entries, we extend our hardness results to any number of centers and large values of threshold, leaving a narrow range of thresholds unresolved.
