Instant Runoff Voting on Graphs: Exclusion Zones and Distortion
This work addresses algorithmic challenges in voting theory for researchers, offering incremental insights into tractability and distortion in structured settings.
The paper tackles the computational complexity of determining exclusion zones in instant-runoff voting on graphs, showing that these problems are NP-hard in general but solvable in polynomial time on trees, and provides bounds on utilitarian distortion for specific graph scenarios.
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.