比例批准投票、调和k中位数和负关联
Proportional Approval Voting, Harmonic k-median, and Negative Association
摘要：
我们研究了一个通用的框架，它提供了对两类重要问题的统一看法：（i）k中值问题的扩展，其中客户对在其附近拥有多个设施感兴趣（例如，由于最近的设施可能以某种小概率发生故障并且因此可能无法使用的事实），以及（i i）根据某些吸引人的多赢家选举规则来寻找赢家，即，一种选举制度，旨在根据选民对个别候选人的偏好来选择代表机构，如议会。在我们的框架中的每个问题都与一个向量的权重：我们表明，问题的可逼近性取决于这些向量的结构属性。我们特别专注于谐波序列的权重，在多赢家选举设置中解释的目标函数反映了众所周知的比例批准投票（PAV）规则。我们的主要结果是，由于特定的（谐波）结构的权重，问题允许常数因子近似。这是令人惊讶的，因为这个问题可以被解释为k-中位数问题的变体，其中我们不假设连接成本满足三角不等式。我们提出的算法是基于依赖舍入[Srinivasan，FOCS'01]适用于自然LP松弛问题的解决方案。众所周知，舍入过程产生满足负相关（NC）的积分解的分布，这通常足以分析舍入过程提供的近似保证。然而，在我们的分析中，我们需要使用这样一个事实，即仔细实现的舍入过程满足一个更强的性质，称为负关联（NA），这允许我们对条件随机变量应用标准的浓度界限。
英文摘要：
We study a generic framework that provides a unified view on two important classes of problems: (i) extensions of the k-median problem where clients are interested in having multiple facilities in their vicinity (e.g., due to the fact that, with some small probability, the closest facility might be malfunctioning and so might not be available for using), and (ii) finding winners according to some appealing multiwinner election rules, i.e., election system aimed for choosing representatives bodies, such as parliaments, based on preferences of a population of voters over individual candidates. Each problem in our framework is associated with a vector of weights: we show that the approximability of the problem depends on structural properties of these vectors. We specifically focus on the harmonic sequence of weights for which the objective function interpreted in a multiwinner election setup reflects to the well-known Proportional Approval Voting (PAV) rule. Our main result is that, due to the specific (harmonic) structure of weights, the problem allows constant factor approximation. This is surprising since the problem can be interpreted as a variant of the k-median problem where we do not assume that the connection costs satisfy the triangle inequality. The algorithm we propose is based on dependent rounding [Srinivasan, FOCS'01] applied to the solution of a natural LP-relaxation of the problem. The rounding process is well known to produce distributions over integral solutions satisfying Negative Correlation (NC), which is usually sufficient for the analysis of approximation guarantees offered by rounding procedures. In our analysis, however, we need to use the fact that the carefully implemented rounding process satisfies a stronger property, called Negative Association (NA), which allows us to apply standard concentration bounds for conditional random variables.
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