The \textsc{Housing Market} problem is a widely studied resource allocation problem. In this problem, each agent can only receive a single object and has preferences over all objects. Starting from an initial endowment, we want to reach a certain assignment via a sequence of rational trades. We first consider whether an object is reachable for a given agent under a social network, where a trade between two agents is allowed if they are neighbors in the network and no participant has a deficit from the trade. Assume that the preferences of the agents are strict (no tie among objects is allowed). This problem is polynomially solvable in a star-network and NP-complete in a tree-network. It is left as a challenging open problem whether the problem is polynomially solvable when the network is a path. We answer this open problem positively by giving a polynomial-time algorithm. Then we show that when the preferences of the agents are weak (ties among objects are allowed), the problem becomes NP-hard even when the network is a path. In addition, we consider the computational complexity of finding different optimal assignments for the problem under the network being a path or a star.

这篇论文在研究房屋市场的资源分配问题，使用社交网络模型，探讨代理人之间的交易，提出了多项解决方案，其中包括了一种可在多项偏好下求解的多项式时间算法以及在路径网络中求解最优分配问题的复杂性分析。

通过严格和弱偏好的交换实现对象可到达性