On the parameterized complexity of manipulating Top Trading Cycles

Abstract: 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}.