Popular matchings in the weighted capacitated house allocation problem

We consider the problem of finding a popular matching in the Weighted Capacitated House Allocation problem (WCHA). An instance of WCHA involves a set of agents and a set of houses. Each agent has a positive weight indicating his priority, and a preference list in which a subset of houses are ranked in strict order. Each house has a capacity that indicates the maximum number of agents who could be matched to it. A matching M of agents to houses is popular   if there is no other matching M′M′ such that the total weight of the agents who prefer their allocation in M′M′ to that in M exceeds the total weight of the agents who prefer their allocation in M   to that in M′M′. Here, we give an O(Cn1+m) algorithm to determine if an instance of WCHA admits a popular matching, and if so, to find a largest such matching, where C   is the total capacity of the houses, n1n1 is the number of agents, and m is the total length of the agents' preference lists.