# On the parameterized complexity of manipulating Top Trading Cycles

## TL;DR

This paper investigates the parameterized complexity of manipulating the Top Trading Cycles mechanism in exchange problems with indivisible goods, revealing polynomial-time algorithms for fixed endowment sizes but also W[1]-hardness results indicating computational difficulty.

## Contribution

It is the first to analyze the parameterized complexity of TTC manipulation, showing both tractable cases and hardness results based on endowment size.

## Key findings

- Manipulation is polynomial-time for fixed endowment sizes.
- The problem is W[1]-hard, indicating unlikely fixed-parameter tractability.
- Results inform the computational limits of strategic manipulation in exchange mechanisms.

## 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}.