AI 中文总结
该研究将计算社会选择从离散智能体拓展至社会连续体设定，针对选举攻击问题证明部分问题可多项式时间求解，部分仍具计算难度，运用了配置线性规划等技术。
AI 中文摘要
计算社会选择旨在为偏好聚合、选举安全性、结果稳健性、稳定性等问题提供算法层面的解答，它大多将社会建模为由离散智能体构成的集合。我们提出在社会连续体（society continuum）设定下研究计算社会选择问题，该设定将社会建模为无穷多不同类型的无穷小微分智能体的分布。类似方法在物理学（是统计力学的基础）、经济学（平均场博弈）等领域已被证明非常有用。作为初始案例研究，我们聚焦于离散设定下已被广泛研究的选举攻击（贿赂与操控）问题。我们表明，在社会连续体设定中，一大类标准选举攻击问题可在多项式时间内求解，该类问题包含离散设定下为NP难的问题，其中包括Borda和Bucklin的CCDV问题、单位成本下的Borda-SWAP贿赂问题。此外，我们针对k为常数且成本为一般成本时的k-Approval-SWAP贿赂问题，以及k可变且成本函数为可加可分时的该问题，给出了多项式时间算法。后一结果与离散问题形成对比，我们证明离散问题在可加可分成本及每个固定k≥2时为NP完全问题。与之相反，我们证明Borda-SWAP贿赂问题和k-Approval-SWAP贿赂问题在一般成本下，于社会连续体设定中仍具有计算难度。为获得这些结果，我们运用了连续与离散优化技术，如配置线性规划（Configuration LP）框架和动态规划。尤其值得注意的是我们难度证明背后的技术，该技术展示了如何在线性规划公式与定价问题之间“反转难度流”。
英文摘要
Computational social choice seeks algorithmic answers to questions about preference aggregation, safety of elections, robustness of outcomes, stability, etc. It overwhelmingly models societies as composed of discrete agents. We propose to study computational social choice problems in a society continuum} setting, where a society is modeled as a distribution of infinitely many infinitesimal agents of different types. An analogous approach has been very useful in physics (it is the basis of statistical mechanics), economics (mean field games), and other fields. As an initial case study, we focus on election attacks (bribery and control), which have been extensively studied in the discrete setting. We show that a broad class of standard election attacks becomes polynomial-time solvable in the society continuum. The class contains problems that are NP-hard discretely, among them Borda- and Bucklin-CCDV and unit-cost Borda-SWAP BRIBERY. Furthermore, we give polynomial-time algorithms for $k$-Approval-SWAP BRIBERY when $k$ is constant for general costs, and when $k$ varies and the cost function is additively separable. The latter result contrasts with the discrete problem, which we prove NP-complete for additively separable costs and every fixed $k\ge 2$. In contrast, we prove that Borda-SWAP BRIBERY and $k$-Approval-SWAP BRIBERY, both with general costs, remain computationally hard in the society continuum. To obtain these results, we use both continuous and discrete optimization techniques, such as the Configuration LP framework and dynamic programming. Of particular note is the technique underlying our hardness proofs, which shows how to ''reverse the flow of hardness'' between LP formulations and pricing problems.
