Instant Runoff Voting on Graphs: Exclusion Zones and Distortion

2026-03-11

AI Summary

This study investigates the elimination zone determination, minimum elimination zone computation, and social welfare distortion under Instant-Runoff Voting (IRV) in graph-induced preference settings. Focusing on unweighted graphs that induce metric preferences, the work proposes a dynamic programming–based Kill membership testing algorithm, achieving the first polynomial-time solution for both elimination zone verification and minimum elimination zone computation on tree structures. The paper further establishes that the corresponding problems are NP-hard or co-NP-complete on general graphs. Additionally, it characterizes the computational complexity of positional elimination rules satisfying the strong compulsory elimination property and derives tight upper and lower bounds on the utilitarian distortion of IRV in perfect binary trees and general unweighted graphs.

Technology Category

• Game Theory and Economic Paradigms: Social Choice / Voting
• Knowledge Representation and Reasoning: Preferences
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Web Mining and Content Analysis: Models for Web evolution

Abstract

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.

Problem

• Instant Runoff Voting
• Exclusion Zones
• Metric Preferences
• Distortion
• Graphs

Innovation

• Instant Runoff Voting
• Exclusion Zones
• Metric Preferences on Graphs
• Dynamic Programming on Trees
• Utilitarian Distortion