Tractable Exclusion Zones for Instant-Runoff Voting on Trees and Beyond
Georgios Birmpas, Georgios Chionas, Efthyvoulos Drousiotis, Soodeh Habibi, Marios Mavronicolas, Paul Spirakis

TL;DR
This paper investigates exclusion zones in instant-runoff voting on trees and general graphs, providing polynomial algorithms for trees and complexity results for general graphs, with implications for robustness certificates.
Contribution
It introduces polynomial-time algorithms for verifying and computing exclusion zones on trees and extends intractability results to broader classes of graphs.
Findings
Exclusion-zone verification is co-NP-complete on general graphs.
On trees, verification and minimum-zone computation are polynomial-time.
Minimum exclusion zones on trees are generated by the closure of a single vertex.
Abstract
Instant-runoff voting (IRV) is often used when voters rank candidates rather than choosing only one favourite. We study IRV under graph-induced metric preferences where each vertex of an unweighted undirected graph hosts one voter and is also a possible candidate location. Voters rank candidates by shortest-path distance with fixed deterministic tie-breaking. We focus on exclusion zones, i.e., sets S such that, whenever at least one candidate lies in S, the IRV winner must also lie in S. Such zones serve as robustness certificates, identifying regions whose participation prevents outside winners from emerging. For general graphs, exclusion-zone verification is co-NP-complete and minimum-zone computation is NP-hard. We show that both problems become polynomial-time solvable on trees. Our main tool is a membership test asking whether a candidate can be forced to lose using opponents from…
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