On randomized approximation for finding a level ideal of a poset and the generalized median stable matchings

This study is concerned with finding a level ideal (LI) of a partially ordered set (poset). Given a finite poset P, the level of each element p ε P is defined as the number of ideals that do not include p, then the problem is to find the ith LI-the ideal consisting of elements whose levels are less than a given integer i. The concept of a level ideal is naturally derived from the generalized median stable matchings, introduced by Teo and Sethuraman [Teo, C. P., J. Sethuraman. 1998. The geometry of fractional stable matchings and its applications. Math. Oper. Res. 23(4) 874-891] in the context of "fairness" of matchings in a stable marriage problem. Cheng [Cheng, C. T. 2010. Understanding the generalized median stable matchings. Algorithmica 58(1) 34-51] showed that finding the ith LI is #P-hard when i = Θ4N5, where N is the total number of ideals of P. This paper shows that finding the ith LI is #P-hard even if i = Θ4N1/c5, where c is an arbitrary constant at least one. Meanwhile, we present a polynomial time exact algorithm when i - O44logN5c0 5, where c0 is an arbitrary positive constant. We also devise two randomized approximation schemes for the ideals of a poset, by using an oracle of an almost-uniform sampler.