[论文解读] Proportional Approval Voting, Harmonic k-median, and Negative Association

Jarosław Byrka, Piotr Skowron+1 more|arXiv (Cornell University)|Apr 7, 2017

Game Theory and Voting Systems参考文献 27被引用 12

一句话总结

本文提出了一种统一框架，用于处理带加权客户成本的OWA k-中位问题，表明调和权重（1, 1/2, 1/3, ..., 1/k）可在不依赖三角不等式的情况下实现常数因子近似——这是设施选址领域的一个令人惊讶的结果。作者基于负相关性（Negative Association）设计了一种相依舍入算法，实现了常数因子近似比，这是首个在无三角不等式假设下针对k-中位变体的此类结果。

ABSTRACT

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.

研究动机与目标

• 将多胜者批准投票与多设施k-中位问题统一到一个共同的OWA k-中位框架下进行研究。
• 基于权重向量的结构特性，分析OWA k-中位问题的可近似性。
• 表明调和权重（1, 1/2, ..., 1/k）具有特别有利的近似性质，可实现常数因子近似。
• 为不假设成本满足三角不等式的调和k-中位问题，设计一种新颖的近似算法。
• 在标准复杂性假设下，为其他权重序列（如p-几何权重）建立强不可近似性结果。

提出的方法

• 使用非递增权重向量来聚合所选设施上的客户成本，从而构建OWA k-中位问题的公式。
• 以OWA k-中位问题的自然线性规划松弛作为舍入过程的起点。
• 对LP解应用相依舍入（Srinivasan, FOCS'01），生成整数设施集合。
• 利用舍入过程中更强的负相关性（NA）性质，对条件随机变量应用集中不等式。
• 证明由于权重的调和结构，舍入过程可实现常数因子近似。
• 通过从精确集合覆盖（X3C）和集合覆盖问题的归约，建立不可近似性结果，表明对一般权重和p-几何权重的困难性。

实验结果

研究问题

• RQ1 在不假设客户-设施成本满足三角不等式的情况下，调和权重的OWA k-中位问题是否可实现常数因子近似？
• RQ2 权重向量的何种结构特性可使OWA k-中位问题实现常数因子近似？
• RQ3 当成本不满足三角不等式时，是否存在调和k-中位问题的多项式时间近似算法？
• RQ4 对于p-几何权重（w_i = p^{i-1}），OWA k-中位问题的可近似性阈值是什么？
• RQ5 该框架能否在多项式时间内近似比例批准投票（PAV）规则？

主要发现

• 即使在不假设成本满足三角不等式的情况下，调和k-中位问题仍存在常数因子近似算法，这是首个针对k-中位变体的此类结果。
• 近似的关键在于调和权重结构，该结构使负相关性在相依舍入中得以应用，从而界定了误差概率。
• 在给定框架下，调和k-中位问题存在时间复杂度为多项式的(1+ε)-近似算法。
• 对于p < 1/e的p-几何权重，除非P = NP，否则不存在常数因子近似算法。
• 对于仅考虑λ分之一设施（λ < 1）的权重向量，OWA k-中位问题在任何正因子内均不可近似，除非P = NP。
• 比例批准投票（PAV）规则（对应于0-1成本与调和权重）被包含在调和k-中位框架中，并继承了相同的近似保证。