We study instant-runoff voting (IRV) under metric preferences induced by an unweighted graph where each vertex hosts a voter, candidates occupy some vertices (with a single candidate allowed in such a vertex), and voters rank candidates by shortest-path distance with fixed deterministic tie-breaking. We focus on exclusion zones, vertex sets S such that whenever some candidate lies in S, the IRV winner must also lie in S. While testing whether a given set S is an exclusion zone is co-NP-Complete and finding the minimum exclusion zone is NP-hard in general graphs, we show here that both problems can be solved in polynomial time on trees. Our approach solves zone testing by designing a Kill membership test (can a designated candidate be forced to lose using opponents from a restricted set?) and shows that Kill can be decided in polynomial time on trees via a bottom-up dynamic program that certifies whether the designated candidate can be eliminated in round 1. A greedy shrinking process then recovers the minimum zone under a standard nesting assumption. To clarify the limits of tractability beyond trees, we also identify a rule level property (Strong Forced Elimination) that abstracts the key IRV behavior used in prior reductions, and show that both exclusion-zone verification and minimum- zone computation remain co-NP-complete and NP-hard, respectively, for any deterministic rank-based elimination rule satisfying this property. Finally, we relate IRV to utilitarian distortion in this discrete setting, and we present upper and lower bounds with regard to the distortion of IRV for several scenarios, including perfect binary trees and unweighted graphs.

翻译：本文研究基于无权图诱导的度量偏好下的即时决选投票机制。在该模型中，每个顶点代表一位选民，候选人占据部分顶点（允许单个顶点仅容纳一位候选人），选民依据最短路径距离对候选人进行排序，并采用固定的确定性平局决胜规则。我们重点关注排除区域——即满足以下性质的顶点集合S：当任意候选人位于S内时，IRV的获胜者也必须位于S内。尽管在一般图中判定给定集合S是否为排除区域属于co-NP完全问题，且寻找最小排除区域属于NP难问题，但本文证明在树结构上这两个问题均存在多项式时间解法。我们的方法通过设计"淘汰成员测试"（能否利用受限对手集合迫使指定候选人落败？）来解决区域判定问题，并证明在树结构上可通过自底向上的动态规划在多项式时间内判定指定候选人能否在第一轮被淘汰，该动态规划可生成候选人淘汰可行性的验证证书。在标准嵌套假设下，通过贪心收缩过程可得到最小区域。为明确树结构之外的可处理性边界，我们还提出了一种规则层级属性（强强制淘汰），该属性抽象了先前归约中使用的IRV关键行为，并证明对于任何满足此属性的确定性基于排序的淘汰规则，排除区域验证与最小区域计算问题分别保持co-NP完全性与NP难性。最后，我们在此离散设定下将IRV与功利主义失真度相关联，针对包括完美二叉树和无权图在内的多种场景，给出了IRV失真度的上下界。