最近的一些作品[Goldberg 2006; O 'Donnell and Servedio 2011; De，Diakonikolas，and Servedio 2017; De，Diakonikolas，Feldman，and Servedio 2014]考虑了在给定各种"权力指数“的信息的情况下近似重建未知加权投票方案的问题，这些”权力指数“表征了个体选民对最终结果的控制水平。在理论计算机科学的语言中，这是在给定一些数值度量（例如函数的$n$的“Chow参数”，也称为“Chow参数”）的情况下，在$\{-1, 1\}^n$上近似未知线性阈值函数（LTF）的问题。它的1阶傅立叶系数，或它的$n$ Shapley指数的向量），每个$n$单独的输入变量对函数的结果有多大影响。在本文中，我们考虑的问题，重建LTF只给出部分信息，其周参数或Shapley指数，即我们只给出周参数或Shapley指数对应的一个子集$S \subseteq [n]$的$n$输入变量。该部分信息设置中的自然目标是找到其Chow参数或对应于$S$中的索引的Shapley索引精确地匹配未知LTF的给定Chow参数或Shapley索引的LTF。我们称之为部分逆幂指数问题。我们的主要结果是（$\varepsilon$ -近似）Chow参数部分逆幂指数问题的多项式时间算法和（$\varepsilon$ -近似）Shapley指数部分逆幂指数问题的拟多项式时间算法.
英文摘要：
A number of recent works [Goldberg 2006; O'Donnell and Servedio 2011; De, Diakonikolas, and Servedio 2017; De, Diakonikolas, Feldman, and Servedio 2014] have considered the problem of approximately reconstructing an unknown weighted voting scheme given information about various sorts of ``power indices'' that characterize the level of control that individual voters have over the final outcome. In the language of theoretical computer science, this is the problem of approximating an unknown linear threshold function (LTF) over $\{-1, 1\}^n$ given some numerical measure (such as the function's $n$ ``Chow parameters,'' a.k.a. its degree-1 Fourier coefficients, or the vector of its $n$ Shapley indices) of how much each of the $n$ individual input variables affects the outcome of the function.
In this paper we consider the problem of reconstructing an LTF given only partial information about its Chow parameters or Shapley indices; i.e. we are given only the Chow parameters or the Shapley indices corresponding to a subset $S \subseteq [n]$ of the $n$ input variables. A natural goal in this partial information setting is to find an LTF whose Chow parameters or Shapley indices corresponding to indices in $S$ accurately match the given Chow parameters or Shapley indices of the unknown LTF. We refer to this as the Partial Inverse Power Index Problem.
Our main results are a polynomial time algorithm for the ($\varepsilon$-approximate) Chow Parameters Partial Inverse Power Index Problem and a quasi-polynomial time algorithm for the ($\varepsilon$-approximate) Shapley Indices Partial Inverse Power Index Problem.
Voting power in the Council of the European Union: A comprehensive sensitivity analysis
Central Bank of Hungary（匈牙利国家银行）；Budapest Metropolitan University, MNB Institute（布达佩斯都会大学，MNB研究所）；Institute for Computer Science and Control (SZTAKI), Hungarian Research Network (HUN-REN)（计算机科学与控制研究所（SZTAKI），匈牙利研究网络（HUN-REN））；Laboratory on Engineering and Management Intelligence, Research Group of Operations Research and Decision Systems（工程与管理智能实验室，运筹学与决策系统研究组）；Corvinus University of Budapest (BCE), Institute of Operations and Decision Sciences, Department of Operations Research and Actuarial Sciences（布达佩斯考文纽斯大学（BCE），运营与决策科学研究所，运筹学与精算科学系）
