The complexity of counting stable marriages
SIAM Journal on Computing
The stable marriage problem: structure and algorithms
The stable marriage problem: structure and algorithms
On the complexity of dynamic programming for sequencing problems with precedence constraints
Annals of Operations Research
Exact sampling with coupled Markov chains and applications to statistical mechanics
Proceedings of the seventh international conference on Random structures and algorithms
The Geometry of Fractional Stable Matchings and its Applications
Mathematics of Operations Research
Searching, Sorting and Randomised Algorithms for Central Elements and Ideal Counting in Posets
Proceedings of the 13th Conference on Foundations of Software Technology and Theoretical Computer Science
Many-to-One Stable Matching: Geometry and Fairness
Mathematics of Operations Research
Sampling stable marriages: why spouse-swapping won't work
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
Finding a Level Ideal of a Poset
COCOON '09 Proceedings of the 15th Annual International Conference on Computing and Combinatorics
Understanding the Generalized Median Stable Matchings
Algorithmica - Special Issue: Matching Under Preferences; Guest Editors: David F. Manlove, Robert W. Irving and Kazuo Iwama
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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. Algorithmica58(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.