  We study multi-type housing markets, where there are $p\ge 2$ types of items,
each agent is initially endowed one item of each type, and the goal is to
design mechanisms without monetary transfer to (re)allocate items to the agents
based on their preferences over bundles of items, such that each agent gets one
item of each type. In sharp contrast to classical housing markets, previous
studies in multi-type housing markets have been hindered by the lack of natural
solution concepts, because the strict core might be empty.
  We break the barrier in the literature by leveraging AI techniques and making
natural assumptions on agents' preferences. We show that when agents'
preferences are lexicographic, even with different importance orders, the
classical top-trading-cycles mechanism can be extended while preserving most of
its nice properties. We also investigate computational complexity of checking
whether an allocation is in the strict core and checking whether the strict
core is empty. Our results convey an encouragingly positive message: it is
possible to design good mechanisms for multi-type housing markets under natural
assumptions on preferences.
