Abstract
In this paper, we are concerned with the generalized popular condensation problem, which is an extension of the popular matching problem. An instance of the popular matching problem consists of a set of applicants \(A\), a set of posts \(P\), and a set of preference lists. According to the preference lists, the goal is to match each applicant with at most one unique post so that the resulting matching is “popular.” A matching \(M\) mapping from \(A\) to \(P\) is popular if there is no other matching \(M'\) such that more applicants prefer \(M'\) to \(M\) than prefer \(M\) to \(M'\). However, such a matching does not always exist. To fulfill the popular matching requirements, a possible manipulation is to neglect some applicants. Given that each applicant is appended with a cost for neglecting, the generalized popular condensation problem asks for a subset of applicants, with minimum sum of costs, whose deletion results in an instance admitting a popular matching. We prove that this problem is NP-hard, and it is also NP-hard to approximate to within a certain factor, assuming ties are involved in the preference lists. For the special case where the costs of all applicants are equal, we show that the problem can be solved in \(O(\sqrt{n}m)\) time, where \(n\) is the number of applicants and posts, and \(m\) is the total length of the preference lists.
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References
Abraham, D.J., Irving, R.W., Kavitha, T., Mehlhorn, K.: Popular matchings. SIAM Journal on Computing 37(4), 1030–1045 (2007)
Abraham, D.J., Kavitha, T.: Dynamic matching markets and voting paths. In: Proceedings of the 10th Scandinavian Workshop on Algorithm Theory, pp. 65–76 (2006)
Biró, P., Irving, R.W., Manlove, D.F.: Popular matchings in the marriage and roommates problems. In: Calamoneri, T., Diaz, J. (eds.) CIAC 2010. LNCS, vol. 6078, pp. 97–108. Springer, Heidelberg (2010)
Gardenfors, P.: Match Making: assignments based on bilateral preferences. Behavioural Sciences 20, 166–173 (1975)
Pulleyblank, W.R.: Matchings and Extensions. In: Graham, R.L., Grötschel, M., Lovasz, L. (eds.): The Handbook of Combinatorics, ch. 3, pp. 179–232. MIT Press, Cambridge (1995)
Huang, C.-C., Kavitha, T.: Near-popular matchings in the roommates problem. SIAM Journal on Discrete Mathematics 27(1), 43–62 (2013)
Huang, C.-C., Kavitha, T.: Popular matchings in the stable marriage problem. Information and Computation 222, 180–194 (2013)
Hopcroft, J.E., Karp, R.M.: A \(n^{5/2}\) algorithm for maximum matchings in bipartite graphs. SIAM Journal on Computing 2(4), 225–231 (1973)
Huang, C.-C., Kavitha, T., Michail, D., Nasre, M.: Bounded unpopularity matching. In: Proceedings of the 12th Scandinavian Workshop on Algorithm Theory, pp. 127–137 (2008)
Kavitha, T.: Popularity vs maximum cardinality in the stable marriage setting. In: Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 123–134 (2012)
Kavitha, T., Nasre, M.: Optimal popular matchings. Discrete Applied Mathematics 157(14), 3181–3186 (2009)
Kavitha, T., Nasre, M.: Popular matchings with variable item copies. Theoretical Computer Science 412, 1263–1274 (2011)
Kavitha, T., Nasre, M., Nimbhorkar, P.: Popularity at minimum cost. In: Cheong, O., Chwa, K.-Y., Park, K. (eds.) ISAAC 2010, Part I. LNCS, vol. 6506, pp. 145–156. Springer, Heidelberg (2010)
Lovasz, L., Plummer, M.D.: Matching Theory. North-Holland (1986)
Mahdian, M.: Random popular matchings. In: Proceedings of the 7th ACM Conference on Electronic Commerce, pp. 238–242 (2006)
Manlove, D.F., Sng, C.T.S.: Popular matchings in the capacitated house allocation problem. In: Proceedings of the 14th Annual European Symposium on Algorithms, pp. 492–503 (2006)
McCutchen, R.M.: The least-unpopularity-factor and least-unpopularity-margin criteria for matching problems with one-sided preferences. In: Laber, E.S., Bornstein, C., Nogueira, L.T., Faria, L. (eds.) LATIN 2008. LNCS, vol. 4957, pp. 593–604. Springer, Heidelberg (2008)
McDermid, E., Irving, R.W.: Popular matchings: structure and algorithms. Journal of Combinatorial Optimization 22(3), 339–358 (2011)
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)
Paluch, K.: Popular and clan-popular \(b\)-matchings. In: Proceedings of the 23rd International Symposium on Algorithms and Computation, pp. 116–125 (2012)
Schaefer, T.J.: The complexity of satisfiability problems. In: Proceedings of the 10th Annual ACM Symposium on Theory of Computing, New York, pp. 216–226 (1978)
Wu, Y.-W., Lin, W.-Y., Wang, H.-L., Chao, K.-M.: An optimal algorithm for the popular condensation problem. In: Lecroq, T., Mouchard, L. (eds.) IWOCA 2013. LNCS, vol. 8288, pp. 412–422. Springer, Heidelberg (2013)
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Wu, YW., Lin, WY., Wang, HL., Chao, KM. (2014). The Generalized Popular Condensation Problem. In: Ahn, HK., Shin, CS. (eds) Algorithms and Computation. ISAAC 2014. Lecture Notes in Computer Science(), vol 8889. Springer, Cham. https://doi.org/10.1007/978-3-319-13075-0_48
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