Abstract
Let I be an instance of the stable marriage (SM) problem. In the late 1990s, Teo and Sethuraman discovered the existence of median stable matchings, which are stable matchings that match all participants to their (lower/upper) median stable partner. About a decade later, Cheng showed that not only are they locally-fair, but they are also globally-fair in the following sense: when G(I) is the cover graph of the distributive lattice of stable matchings, these stable matchings are also medians of G(I) – i.e., their average distance to the other stable matchings is as small as possible. Unfortunately, finding a median stable matching of I is #P-hard.
Inspired by the fairness properties of the median stable matchings, we study the center stable matchings which are the centers of G(I) – i.e., the stable matchings whose maximum distance to any stable matching is as small as possible. Here are our two main results. First, we show that a center stable matching of I can be computed in O(|I|2.5) time. Thus, center stable matchings are the first type of globally-fair stable matchings we know of that can be computed efficiently. Second, we show that in spite of the first result, there are similarities between the set of median stable matchings and the set of center stable matchings of I. The former induces a hypercube in G(I) while the latter is the union of hypercubes of a fixed dimension in G(I). Furthermore, center stable matchings have a property that approximates the one found by Teo and Sethuraman for median stable matchings. Finally, we note that our results extend to other variants of SM whose solutions form a distributive lattice and whose rotation posets can be constructed efficiently.
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References
Abdulkadiroglu, A., Pathak, P., Roth, A.: The New York City high school match. American Economic Review, Papers and Proceedings 95, 364–367 (2005)
Abdulkadiroglu, A., Pathak, P., Roth, A., Sönmez, T.: The Boston public school match. American Economic Review, Papers and Proceedings 95, 368–371 (2005)
Barbut, M.: Médiane, distributivité, éloignements, 1961. Reprinted in. Mathématiques et Sciences Humaines 70, 5–31 (1980)
Bhatnagar, N., Greenberg, S., Randall, D.: Sampling stable marriages: why spouse-swapping won’t work. In: Proc. of SODA 2008, pp. 1223–1232 (2008)
Birkhoff, G.: Rings of sets. Duke Mathematical Journal 3, 443–454 (1937)
Blair, C.: Every finite distributive lattice is a set of stable matchings. J. Combin. Theory, Ser. A 37, 353–356 (1984)
Chebolu, P., Goldberg, L., Martin, R.: The complexity of approximately counting stable matchings. In: Serna, M., Shaltiel, R., Jansen, K., Rolim, J. (eds.) APPROX 2010, LNCS, vol. 6302, pp. 81–94. Springer, Heidelberg (2010)
Cheng, C.: Understanding the generalized median stable matchings. Algorithmica 58, 34–51 (2010)
Chepoi, V., Dragan, F., Vaxéx, Y.: Center and diameter problems in plane triangulations and quadrangulations. In: Proc. of SODA 2002, pp. 346–355 (2002)
Felsner, S.: Lattice structure from planar graphs. Electronic Journal of Combinatorics 11, R15 (2004)
Gale, D., Shapley, L.: College admissions and the stability of marriage. American Mathematical Monthly 69 (1962)
Gusfield, D.: Three fast algorithms for four problems in stable marriage. SIAM Journal on Computing 16, 111–128 (1987)
Gusfield, D., Irving, R.: The Stable Marriage Problem: Structure and Algorithms. The MIT Press, Cambridge (1989)
Imrich, W., Klavžar, S.: Product Graphs: Structure and Recognition. Wiley Interscience, Hoboken (2000)
Irving, R., Leather, P., Gusfield, D.: An efficient algorithm for the optimal stable marriage. Journal of the ACM 34, 532–544 (1987)
Knuth, D.: Mariages Stables. Les Presses de l’Université de Montréal (1976)
König, D.: Gráfokés mátrixok. Matematikaiés Fizikai Lapok 38, 116–119 (1931)
Micali, S., Vazirani, V.: An \({O(\sqrt{V}E})\) algorithm for finding maximum matchings in general graphs. In: Proc. of FOCS 1980, pp. 17–27 (1980)
Nieminen, J.: Distance center and centroid of a median graph. Journal of the Franklin Institute 323, 89–94 (1987)
Propp, J.: Generating random elements of finite distributive lattices. Electronic Journal of Combinatorics 4 (1997)
Roth, A., Peranson, E.: The redesign of the matching market of American physicians: Some engineering aspects of economic design. American Economic Review 89, 748–780 (1999)
Teo, C.-P., Sethuraman, J.: The geometry of fractional stable matchings and its applications. Mathematics of Operations Research 23, 874–891 (1998)
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Cheng, C., McDermid, E., Suzuki, I. (2011). Center Stable Matchings and Centers of Cover Graphs of Distributive Lattices. In: Aceto, L., Henzinger, M., Sgall, J. (eds) Automata, Languages and Programming. ICALP 2011. Lecture Notes in Computer Science, vol 6755. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22006-7_57
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DOI: https://doi.org/10.1007/978-3-642-22006-7_57
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