操纵最高交易周期的参数化复杂性
On the parameterized complexity of manipulating Top Trading Cycles
摘要：
我们研究的交换问题时，1）代理人被赋予异构不可分割的对象，2）没有钱。一般来说，没有规则满足帕累托效率、个体理性和防策略性\cite{Sonmez1999}这三个核心属性。最近的研究表明，最佳交易周期是$\NP$ --很难操纵的\cite{FujitaEA2015}，这是对策略防御性的放松。然而，参数化的复杂性是一个更合适的框架，这种经济环境和其他经济环境。问题的某些方面--每个代理人带到桌面上的对象数量、拍卖的货物、选举\cite{consandlang2007}中的候选人、影响\cite{christian2007complexity}的立法人物--可能面临自然界限，或者随着问题的发展而固定。首次对不可分商品交换问题进行了参数化复杂性分析。我们的结果对TTC来说既有好消息，也有坏消息。当禀赋的大小$k$是一个固定的常数，我们表明，操作TTC的计算任务可以在多项式时间。另一方面，我们表明，这个参数化的问题是$\W[1]$ -硬，因此不太可能是\emph{fixed parameter tractable}。
英文摘要：
We study the problem of exchange when 1) agents are endowed with heterogeneous indivisible objects, and 2) there is no money. In general, no rule satisfies the three central properties Pareto-efficiency, individual rationality, and strategy-proofness \cite{Sonmez1999}. Recently, it was shown that Top Trading Cycles is $\NP$-hard to manipulate \cite{FujitaEA2015}, a relaxation of strategy-proofness. However, parameterized complexity is a more appropriate framework for this and other economic settings. Certain aspects of the problem - number of objects each agent brings to the table, goods up for auction, candidates in an election \cite{consandlang2007}, legislative figures to influence \cite{christian2007complexity} - may face natural bounds or are fixed as the problem grows. We take a parameterized complexity approach to indivisible goods exchange for the first time. Our results represent good and bad news for TTC. When the size of the endowments $k$ is a fixed constant, we show that the computational task of manipulating TTC can be performed in polynomial time. On the other hand, we show that this parameterized problem is $\W[1]$-hard, and therefore unlikely to be \emph{fixed parameter tractable}.
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