We first consider the problem of finding a maximum stable matching if incomplete lists and ties are both allowed, but ties only for one gender. 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 [5]. 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 [7]. For the cases considered in this paper it is NP-hard to approximate within a factor of 21/19 by the result of Halldorsson et al. [3]. 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 [5] and the analysis is substantially more moderate.
展开▼
