[Paper Review] Instant Runoff Voting on Graphs: Exclusion Zones and Distortion

Georgios Birmpas, Georgios Chionas+4 more|arXiv (Cornell University)|Mar 11, 2026

Game Theory and Voting Systems0 citations

TL;DR

The paper studies IRV on metric preferences induced by unweighted graphs, proves polynomial-time exclusion-zone computations on trees via a Kill membership test and dynamic programming, demonstrates hardness beyond trees, and provides distortion bounds for IRV in various unweighted graph scenarios.

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.

Motivation & Objective

• Motivate and formalize IRV under graph-induced metric preferences with deterministic tie-breaking.
• Characterize exclusion zones and develop efficient algorithms for trees.
• Explore rule-level hardness beyond IRV and identify properties that generalize to other elimination rules.
• Analyze distortion of IRV in unweighted graphs, including special tree structures, and establish bounds.

Proposed method

• Define Kill(T,u,A) to test if a designated candidate u can be eliminated using opponents from A.
• Develop a bottom-up dynamic program on trees to decide Kill in polynomial time.
• Use Kill to compute the minimum exclusion zone via a greedy shrinking procedure under nesting.
• Introduce Strong Forced Elimination (SFE) to abstract rule-level invariance and prove co-NP-completeness and NP-hardness results for general graphs with deterministic rank-based rules.
• Analyze distortion by comparing IRV outcomes to social cost in discrete graph metrics and derive upper/lower bounds for paths, bistars, perfect binary trees, and general unweighted graphs.

Experimental results

Research questions

• RQ1 Can we efficiently test whether a given set is an exclusion zone for IRV on graphs, and can we compute the minimum exclusion zone?
• RQ2 Does the tree structure enable tractable (polynomial-time) solutions for exclusion zones and minimum zones under IRV?
• RQ3 How does IRV perform in terms of social cost distortion in unweighted graph metrics, and what are tight bounds for specific tree families and general graphs?
• RQ4 To what extent do hardness results extend to other deterministic rank-based elimination rules that satisfy invariance properties?
• RQ5 What is the relationship between exclusion zones and distortion in graph-based metric voting?

Key findings

• Exclusion-zone verification and minimum-zone computation are tractable in polynomial time on trees using a Kill membership test and a bottom-up DP.
• A round-1 reduction (Kill implies immediate elimination) enables compact DP representations and a greedy shrinking procedure to obtain the minimum exclusion zone under nesting.
• Introducing Strong Forced Elimination (SFE) shows that exclusion-zone verification is co-NP-complete and minimum-zone computation is NP-hard for any deterministic rank-based elimination rule satisfying SFE, extending hardness beyond IRV.
• IRV distortion in unweighted paths is at most 2, with a lower bound of 9/5; bistars yield distortion at most 5/3 (tight for IRV); perfect binary trees yield distortion at most 3 (with IRV not always better than 1.7).
• In general unweighted graphs, IRV distortion is bounded below by Omega(sqrt(log m)) and above by O(log m).