Abstract
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:
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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.
Work done as part of the DST-MPG partner group “Efficient Graph Algorithms” at IISc Bangalore.
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References
Abraham, D.J., Cechlárová, K., Manlove, D.F., Mehlhorn, K.: Pareto-optimality in house allocation problems. In: Fleischer, R., Trippen, G. (eds.) ISAAC 2004. LNCS, vol. 3341, pp. 3–15. Springer, Heidelberg (2004)
Abraham, D.J., Irving, R.W., Kavitha, T., Mehlhorn, K.: Popular matchings. SIAM Journal on Computing 37(4), 1030–1045 (2007)
Abdulkadiroǧluand, A., Sönmez, T.: Random serial dictatorship and the core from random endowments in house allocation problems. Econometrica 66(3), 689–701 (1998)
Denman, R., Foster, S.: Using clausal graphs to determine the computational complexity of k-bounded positive one-in-three SAT. Discrete Applied Mathematics 157(7), 1655–1659 (2009)
Gardenfors, P.: Match making: assignments based on bilateral preferences. Behavioural Sciences 20, 166–173 (1975)
Huang, C.-C., Kavitha, T., Michail, D., Nasre, M.: Bounded unpopularity matchings. In: Gudmundsson, J. (ed.) SWAT 2008. LNCS, vol. 5124, pp. 127–137. Springer, Heidelberg (2008)
Irving, R.W., Kavitha, T., Mehlhorn, K., Michail, D., Paluch, K.: Rank-maximal matchings. ACM Transactions on Algorithms 2(4), 602–610 (2006)
Kavitha, T., Mestre, J., Nasre, M.: Popular Mixed Matchings. In: Albers, S., Marchetti-Spaccamela, A., Matias, Y., Niko-letsea, S. (eds.) ICALP 2009. LNCS, vol. 5556. Springer, Heidelberg (2009)
Mahdian, M.: Random popular matchings. In: Proceedings of the 8th ACM Conference on Electronic Commerce, pp. 238–242 (2006)
Manlove, D., Sng, C.: Popular matchings in the capacitated house allocation problem. In: Azar, Y., Erlebach, T. (eds.) ESA 2006. LNCS, vol. 4168, pp. 492–503. Springer, Heidelberg (2006)
McCutchen, R.M.: The least-unpopularity-factor and least-unpopularity-margin criteria for matching problems with one-sided preferences. In: Proceedings of the 15th Latin American Symposium on Theoretical Informatics, pp. 593–604 (2008)
Mestre, J.: Weighted popular matchings. In: Bugliesi, M., Preneel, B., Sassone, V., Wegener, I. (eds.) ICALP 2006. LNCS, vol. 4051, pp. 715–726. Springer, Heidelberg (2006)
Roth, A.E., Postlewaite, A.: Weak versus strong domination in a market with indivisible goods. Journal of Mathematical Economics 4, 131–137 (1977)
Schaefer, T.J.: The complexity of satisfiability problems. In: Proceedings of the 10th Annual ACM Symposium on Theory of Computing, pp. 216–226 (1978)
Kavitha, T., Nasre, M.: Popular matchings with variable job capacities. Indian Institute of Science. Technical Report. IISc-CSA-TR-2009-7
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Kavitha, T., Nasre, M. (2009). Popular Matchings with Variable Job Capacities. In: Dong, Y., Du, DZ., Ibarra, O. (eds) Algorithms and Computation. ISAAC 2009. Lecture Notes in Computer Science, vol 5878. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-10631-6_44
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DOI: https://doi.org/10.1007/978-3-642-10631-6_44
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