Polynomial-Time Algorithm for Thiele Voting Rules with Voter Interval Preferences

2026-04-07

AI Summary

This work resolves a decade-old open problem concerning the optimal selection of a k-member committee under Thiele voting rules when voters exhibit interval preferences—i.e., each voter approves a contiguous set of candidates. By establishing a concavity theorem for interval families, the authors prove that the optimal Thiele score is concave with respect to committee size. Leveraging this property together with Lagrangian relaxation, total unimodularity, and concave optimization techniques, they devise the first polynomial-time algorithm for this setting. The approach extends to generalized Thiele rules with individual voter weights, thereby enabling efficient exact optimization for all Thiele rules under interval preferences—a challenge explicitly posed in IJCAI 2015 and AAAI 2018.

Abstract

We present a polynomial-time algorithm for computing an optimal committee of size $k$ under any given Thiele voting rule for elections on the Voter Interval domain (i.e., when voters can be ordered so that each candidate is approved by a consecutive voters). Our result extends to the Generalized Thiele rule, in which each voter has an individual weight (scoring) sequence. This resolves a 10-year-old open problem that was originally posed for Proportional Approval Voting and later extended to every Thiele rule (Elkind and Lackner, IJCAI 2015; Peters, AAAI 2018). Our main technical ingredient is a new structural result -- a concavity theorem for families of intervals. It shows that, given two solutions of different sizes, one can construct a solution of any intermediate size whose score is at least the corresponding linear interpolation of the two scores. As a consequence, on Voter Interval profiles, the optimal total Thiele score is a concave function of the committee size. We exploit this concavity within an optimization framework based on a Lagrangian relaxation of a natural integer linear program formulation, obtained by moving the cardinality constraint into the objective. On Voter Interval profiles, the resulting constraint matrix is totally unimodular, so it can be solved in polynomial time. Our main algorithm and its proof were obtained via human--AI collaboration. In particular, a slightly simplified version of the main structural theorem used by the algorithm was obtained in a single call to Gemini Deep Think.

Problem

• Thiele voting rules
• Voter Interval Preferences
• Proportional Approval Voting
• optimal committee selection
• computational social choice

Innovation

• Thiele voting rules
• Voter Interval preferences
• polynomial-time algorithm
• concavity theorem
• Lagrangian relaxation