The 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 polynomial-time solvable in a star-network and NP-complete in a tree-network. It is left as a challenging open problem whether the problem is polynomial-time 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 when the network is a path and can be solved in polynomial time when the network is a star. Besides, we consider the computational complexity of finding different optimal assignments for the problem in the special case where the network is a path or a star.

中文翻译：

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在严格和弱偏好下通过交换实现对象可到达性

在住房市场问题是一个广泛研究的资源分配问题。在此问题中，每个代理只能接收一个对象，并且对所有对象都具有优先级。从最初的end赋开始，我们希望通过一系列理性交易达到一定的任务。我们首先考虑在社交网络下给定代理是否可以到达对象，如果两个代理是网络中的邻居并且没有参与者出现交易赤字，则允许两个代理之间进行交易。假定代理的首选项严格（不允许在对象之间建立联系）。这个问题在星形网络中是多项式时间可解决的，而在树状网络中是NP完全的。当网络是路径时，该问题是否是多项式时间可解决的问题仍然是一个具有挑战性的开放问题。我们通过给出多项式时间算法来肯定地回答这个开放问题。然后我们证明，当代理人的偏好较弱（允许对象之间的联系）时，当网络是路径时，该问题就变得很困难，而当网络是星形时，可以在多项式时间内解决。此外，在网络是路径或星形的特殊情况下，我们考虑为问题找到不同的最佳分配的计算复杂性。