Abstract
The stable marriage problem has received considerable attention both due to its practical applications as well as its mathematical structure. While the original problem has all participants rank all members of the opposite sex in a strict order of preference, two natural variations are to allow for incomplete preference lists and ties in the preferences. Both variations are polynomially solvable by a variation of the classical algorithm of Gale and Shapley. On the other hand, it has recently been shown to be NP-hard to .nd a maximum cardinality stable matching when both of the variations are allowed.
We show here that it is APX-hard to approximate the maximum cardinality stable matching with incomplete lists and ties. This holds for some very restricted instances both in terms of lengths of preference lists, and lengths and occurrences of ties in the lists. We also obtain an optimal Ω(N) hardness results for ‘minimum egalitarian’ and ‘minimum regret’ variants.
Supported in part by Scientific Research Grant, Ministry of Japan, 13480081
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References
S. Arora and C. Lund, “Hardness of Approximations,” Chapter in the bookApproximation Algorithms for NP-hard problems, D. Hochbaum editor, PWS Publishing, 1996.
D. Gale and L. S. Shapley, “College admissions and the stability of marriage,” Amer. Math. Monthly, Vol.69, pp.9–15, 1962.
D. Gale and M. Sotomayor, “Some remarks on the stable matching problem,” Discrete Applied Mathematics, Vol.11, pp.223–232, 1985.
D. Gusfield, “Three fast algorithms for four problems in stable marriage,” SIAM Journal on Computing, Vol. 16, pp. 111–128, 1987.
D. Gusfield and R. W. Irving, “The Stable Marriage Problem: Structure and Algorithms,” MIT Press, Boston, MA, 1989.
J. Håstad, “Some optimal inapproximability results,” Proc. STOC 97, pp. 1–11, 1997.
R. W. Irving, P. Leather and D. Gusfield, “An efficient algorithm for the “optimal” stable marriage,” Journal of the A.C.M., Vol. 34, pp. 532–543, 1987.
R. W. Irving, “Stable marriage and indifference,” Discrete Applied Mathematics, Vol.48, pp.261–272, 1994.
R. W. Irving, D. F. Manlove and S. Scott, “The hospital/residents problem with ties,” Proc. SWAT 2000, LNCS 1851, pp. 259–271, 2000.
K. Iwama, D. Manlove, S. Miyazaki, and Y. Morita, “Stable marriage with incomplete lists and ties,” Proc. ICALP’99, LNCS 1644, pp. 443–452, 1999.
D. Manlove, R. W. Irving, K. Iwama, S. Miyazaki, Y. Morita, “Hard variants of stable marriage,” Technical Report TR-1999-43, Computing Science Department of Glasgow University, September 1999 (to appear in Theoretical Computer Science).
E. Ronn,“NP-complete stable matching problems,” J. Algorithms, Vol.11, pp.285–304, 1990.
T. J. Schaefer, “The complexity of satis.ability problems,” Proc. STOC 78, pp.216–226, 1978.
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Halldórsson, M., Iwama, K., Miyazaki, S., Morita, Y. (2002). Inapproximability Results on Stable Marriage Problems. In: Rajsbaum, S. (eds) LATIN 2002: Theoretical Informatics. LATIN 2002. Lecture Notes in Computer Science, vol 2286. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45995-2_48
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DOI: https://doi.org/10.1007/3-540-45995-2_48
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