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.

翻译：我们提出了一种多项式时间算法，用于在选民区间域（即当选民可以排序使得每个候选人获得连续选民支持时）的选举中，在任意给定Thiele投票规则下计算大小为$k$的最优委员会。该结果可推广至广义Thiele规则，其中每位选民具有独立的权重（计分）序列。这解决了一个持续10年的开放问题，该问题最初针对比例批准投票提出，后扩展至所有Thiele规则（Elkind和Lackner, IJCAI 2015; Peters, AAAI 2018）。我们的主要技术贡献是新的结构性结果——区间族的凹性定理。该定理表明，给定两个不同大小的解，可以构造出任意中间大小的解，其得分至少等于这两个得分相应的线性插值。因此，在选民区间轮廓上，最优总Thiele得分是委员会大小的凹函数。我们利用这种凹性，在一个基于自然整数线性规划形式化的拉格朗日松弛的优化框架中，通过将基数约束移入目标函数来实现。在选民区间轮廓上，由此产生的约束矩阵是完全幺模的，因此可以在多项式时间内求解。我们的主要算法及其证明是通过人机协作获得的。特别是，算法所使用的主要结构定理的略微简化版本是通过单次调用Gemini Deep Think得到的。