The complexity of satisfiability problems
STOC '78 Proceedings of the tenth annual ACM symposium on Theory of computing
EC '06 Proceedings of the 7th ACM conference on Electronic commerce
ACM Transactions on Algorithms (TALG)
Popular matchings in the capacitated house allocation problem
ESA'06 Proceedings of the 14th conference on Annual European Symposium - Volume 14
SIAM Journal on Computing
Bounded Unpopularity Matchings
SWAT '08 Proceedings of the 11th Scandinavian workshop on Algorithm Theory
Discrete Applied Mathematics
ICALP '09 Proceedings of the 36th International Colloquium on Automata, Languages and Programming: Part I
LATIN'08 Proceedings of the 8th Latin American conference on Theoretical informatics
ICALP'06 Proceedings of the 33rd international conference on Automata, Languages and Programming - Volume Part I
Pareto optimality in house allocation problems
ISAAC'04 Proceedings of the 15th international conference on Algorithms and Computation
Popular matchings with variable item copies
Theoretical Computer Science
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We consider the problem of matching people to jobs, where each person ranks a subset of jobs in an order of preference, possibly involving ties. There are several notions of optimality about how to best match each person to a job; in particular, popularity is a natural and appealing notion of optimality. However, popular matchings do not always provide an answer to the problem of determining an optimal matching since there are simple instances that do not admit popular matchings. This motivates the following extension of the popular matchings problem: Given a graph $G = ({\mathcal{A}}\cup{\mathcal{J}},E)$ where ${\mathcal{A}}$ is the set of people and ${\mathcal{J}}$ is the set of jobs, and a list $\langle c_1,\ldots,c_{|{\mathcal{J}}|}\rangle$ denoting upper bounds on the capacities of each job, does there exist $(x_1,\ldots,x_{|{\mathcal{J}}|})$ such that setting the capacity of i-th job to x i , where 1 ≤ x i ≤ c i , for each i, enables the resulting graph to admit a popular matching. In this paper we show that the above problem is NP-hard. We show that the problem is NP-hard even when each c i is 1 or 2.