On Randomized Approximation for Finding a Level Ideal of a Poset and the Generalized Median Stable Matchings

Abstract

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 ∈ 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 = Θ(N), where N is the total number of ideals of P. This paper shows that finding the ith LI is #P-hard even if i = Θ(N1/c), where c is an arbitrary constant at least one. Meanwhile, we present a polynomial time exact algorithm when i = O((log N)c′), where c′ 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.