基于谱技术的简约机制设计
Frugal Mechanism Design via Spectral Techniques
摘要：
我们研究的设计真实的机制，为集系统，即，客户需要雇用一组代理来执行复杂任务的场景。在这种情况下，节俭[Archer&Tardos'02]提供了一个衡量“真实性成本”的标准，即相对于“公平”支付的真实机制的超额支付。我们提出了一个统一的计划，设计节俭的真实机制一般集系统。我们的计划是基于缩放代理人的出价使用的矩阵，编码代理之间的相互依赖性的特征向量。我们证明了r-出-k-系统机制和\sqrt -机制购买图中的路径[Karlin et. 05]可以被看作是我们方案的实例。然后，我们将我们的计划，以其他两类集系统，即顶点覆盖系统和k-路径系统，其中客户需要购买k边不相交的源-汇路径。对于这两种设置，我们约束我们的机制在各自的相互依赖矩阵的最大特征值方面的节俭。我们表明，我们的机制是最佳的一个大子类的顶点覆盖系统满足一个简单的局部稀疏性条件。对于k-路径系统，虽然我们的机制是在一个因素的k + 1从最佳的，我们表明，它是，事实上，最佳的，当一个人使用一个修改后的定义节俭提出[Elkind等人。'07]。我们的下界论证结合了谱技术和Young不等式，并且适用于所有的集合系统。由于r-out-of-k系统和单路系统都可以看作是k-路系统的特殊情况，我们的结果改进了[Karlin et al. [05]并回答了该论文中提出的几个开放性问题。
英文摘要：
We study the design of truthful mechanisms for set systems, i.e., scenarios where a customer needs to hire a team of agents to perform a complex task. In this setting, frugality [Archer&Tardos'02] provides a measure to evaluate the "cost of truthfulness", that is, the overpayment of a truthful mechanism relative to the "fair" payment. We propose a uniform scheme for designing frugal truthful mechanisms for general set systems. Our scheme is based on scaling the agents' bids using the eigenvector of a matrix that encodes the interdependencies between the agents. We demonstrate that the r-out-of-k-system mechanism and the \sqrt-mechanism for buying a path in a graph [Karlin et. al'05] can be viewed as instantiations of our scheme. We then apply our scheme to two other classes of set systems, namely, vertex cover systems and k-path systems, in which a customer needs to purchase k edge-disjoint source-sink paths. For both settings, we bound the frugality of our mechanism in terms of the largest eigenvalue of the respective interdependency matrix. We show that our mechanism is optimal for a large subclass of vertex cover systems satisfying a simple local sparsity condition. For k-path systems, while our mechanism is within a factor of k + 1 from optimal, we show that it is, in fact, optimal, when one uses a modified definition of frugality proposed in [Elkind et al.'07]. Our lower bound argument combines spectral techniques and Young's inequality, and is applicable to all set systems. As both r-out-of-k systems and single path systems can be viewed as special cases of k-path systems, our result improves the lower bounds of [Karlin et al.'05] and answers several open questions proposed in that paper.
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