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. We then combine this Kill-based characterization with additional structural arguments to obtain polynomial-time minimum-zone computation on trees. 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：我们研究在由无权重图诱导的度量偏好下的即时复选投票（IRV），其中每个顶点容纳一名选民，候选者占据某些顶点（同一顶点只允许一名候选者），选民根据最短路径距离，在固定确定性破平规则下对候选者进行排序。我们重点关注排除区，即顶点集 $S$，使得当某候选者位于 $S$ 中时，IRV胜出者必也在 $S$ 中。尽管检验给定集合 $S$ 是否为排除区是co-NP完全的，且在一般图中寻找最小排除区是NP难的，我们在此证明，在树上这两个问题均可在多项式时间内求解。我们的方法通过设计“淘汰”成员测试（能否使用来自限制集合的对手迫使指定候选者失败？）来求解区域检验，并表明在树上可通过自底向上的动态规划在多项式时间内判定“淘汰”：该规划认证指定候选者能否在第一轮被淘汰。随后，我们将此基于“淘汰”的表征与额外结构性论证相结合，得到树上的多项式时间最小区域计算。为厘清树以外可解性的边界，我们还识别了一个规则级性质（强强制淘汰），该性质抽象了先前归约中使用的IRV关键行为，并证明任何满足此性质的确定性逐轮淘汰规则下，排除区验证和最小区域计算分别仍保持co-NP完全性和NP难性。最后，我们在此离散设定下将IRV与功利畸变相联系，并针对若干场景（包括完美二叉树和无权重图）给出IRV畸变的上界和下界。