TAMC'11 Proceedings of the 8th annual conference on Theory and applications of models of computation
Popularity vs maximum cardinality in the stable marriage setting
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
Journal of Discrete Algorithms
Faster and simpler approximation of stable matchings
WAOA'11 Proceedings of the 9th international conference on Approximation and Online Algorithms
Complementary cooperation, minimal winning coalitions, and power indices
Theoretical Computer Science
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We first consider the problem of finding a maximum size stable matching if incomplete lists and ties are both allowed, but ties are on one side only. For this problem we give a simple, linear time 3/2-approximation algorithm, improving on the best known approximation factor 5/3 of Irving and Manlove (J. Comb. Optim., doi: 10.1007/s10878-007-9133-x, 2007). Next, we show how this extends to the Hospitals/Residents problem with the same ratio if the residents have strict orders. We also give a simple linear time algorithm for the general problem with approximation factor 5/3, improving the best known 15/8-approximation algorithm of Iwama, Miyazaki and Yamauchi (SODA ’07: Proceedings of the Eighteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 288–297, 2007). For the cases considered in this paper it is NP-hard to approximate within a factor of 21/19 by the result of Halldórsson et al. (ACM Transactions on Algorithms 3(3):30, 2007). Our algorithms not only give better approximation ratios than the cited ones, but are much simpler and run significantly faster. Also we may drop a restriction used in (J. Comb. Optim., doi: 10.1007/s10878-007-9133-x, 2007) and the analysis is substantially more moderate. Preliminary versions of this paper appeared in (Király, Egres Technical Report TR-2008-04, www.cs.elte.hu/egres/, 2008; Király in Proceedings of MATCH-UP 2008: Matching Under Preferences—Algorithms and Complexity, Satellite Workshop of ICALP, July 6, 2008, Reykjavík, Iceland, pp. 36–45, 2008; Király in ESA 2008, Lecture Notes in Computer Science, vol. 5193, pp. 623–634, 2008). For the related results obtained thenceforth see Sect. 5.