Polynomial-Time Algorithm for Thiele Voting Rules on Voter Interval Domain
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., voters can be ordered so that each candidate is approved by a contiguous block of 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.
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.
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.
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