Abstract
This paper is concerned with finding a level ideal (LI) of a partially ordered set (poset): given a finite poset P, a level of each element p ∈ P is defined as the number of ideals which do not include p, then the problem is to find an ideal consisting of elements whose levels are less than a given integer i. We call the ideal as the i-th LI. The concept of the level ideal is naturally derived from the generalized median stable matching, that is a fair stable marriage introduced by Teo and Sethuraman (1998). Cheng (2008) showed that finding the i-th LI is #P-hard when i = Θ(N), where N is the total number of ideals of P. This paper shows that finding the i-th LI is #P-hard even if i = Θ(N1/c) where c ≥ 1 is an arbitrary constant. Meanwhile, we give a polynomial time exact algorithm when i = O((logN)c′) where c′ is an arbitrary positive constant. We also devise two randomized approximation schemes using an oracle of almost uniform sampler for ideals of a poset.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bhatnagar, N., Greenberg, S., Randall, D.: Sampling stable marriages: why the spouse-swapping won’t work. In: Proc. of SODA 2008, pp. 1223–1232 (2008)
Blair, C.: Every finite distributive lattice is a set of stable matchings. J. Comb. Theory A 37, 353–356 (1984)
Cheng, C.T.: The generalized median stable matchings: finding them is not that easy. In: Proc. of Latin 2008, pp. 568–579 (2008)
Davey, B.A., Priestley, H.A.: Introduction to Lattices and Order, 2nd edn. Cambridge University Press, Cambridge (2002)
Dubhashi, D.P., Mehlhorn, K., Rajan, D., Thiel, C.: Searching, sorting and randomised algorithms for central elements and ideal counting in posets. In: Shyamasundar, R.K. (ed.) FSTTCS 1993. LNCS, vol. 761, pp. 436–443. Springer, Heidelberg (1993)
Gale, D., Shapley, L.S.: College admissions and the stability of marriage. Am. Math. Month. 69, 9–15 (1962)
Gusfield, D., Irving, R.W.: The Stable Marriage Problem, Structure and Algorithms. MIT Press, Cambridge (1989)
Irving, R.W., Leather, P.: The complexity of counting stable marriages. SIAM J. Comput. 15, 655–667 (1986)
Jerrum, M.: Counting, Sampling and Integrating: Algorithms and Complexity. ETH Zürich, Birkhauser, Basel (2003)
Kijima, S., Nemoto, T.: Randomized approximation for generalized median stable matching. RIMS-preprint 1648 (2008)
Klaus, B., Klijn, F.: Median stable matching for college admission. Int. J. Game Theory 34, 1–11 (2006)
Klaus, B., Klijn, F.: Smith and Rawls share a room: stability and medians. Meteor RM/08-009, Maastricht University (2008), http://edocs.ub.unimaas.nl/loader/file.asp?id=1307
Knuth, D.: Stable Marriage and Its Relation to Other Combinatorial Problems. American Mathematical Society (1991)
Propp, J., Wilson, D.B.: Exact sampling with coupled Markov chains and applications to statistical mechanics. Random Struct. Algo. 9, 223–252 (1996)
Nemoto, T.: Some remarks on the median stable marriage problem. In: ISMP 2000 (2000)
Provan, J.S., Ball, M.O.: The complexity of counting cuts and of computing the probability that a graph is connected. SIAM J. Comput. 12, 777–788 (1983)
Roth, A.E., Sotomayor, M.A.O.: A Two-Sided Matchings: A Study In Game-Theoretic Modeling And Analysis. Cambridge University Press, Cambridge (1990)
Sethuraman, J., Teo, C.P., Qian, L.: Many-to one stable matching: geometry and fairness. Math. Oper. Res. 31, 581–596 (2006)
Schwarz, M., Yenmez, M.B.: Median stable matching. NBER Working Paper No. w14689 (2009)
Squire, M.B.: Enumerating the ideals of a poset. preprint, North Carolina State University (1995)
Steiner, G.: An algorithm for generating the ideals of a partial order. Oper. Res. Lett. 5, 317–320 (1986)
Steiner, G.: On the complexity of dynamic programming for sequencing problems with precedence constraints. Ann. Oper. Res. 26, 103–123 (1990)
Teo, C.P., Sethuraman, J.: The geometry of fractional stable matchings and its applications. Math. Oper. Res. 23, 874–891 (1998)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Kijima, S., Nemoto, T. (2009). Finding a Level Ideal of a Poset. In: Ngo, H.Q. (eds) Computing and Combinatorics. COCOON 2009. Lecture Notes in Computer Science, vol 5609. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02882-3_32
Download citation
DOI: https://doi.org/10.1007/978-3-642-02882-3_32
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-02881-6
Online ISBN: 978-3-642-02882-3
eBook Packages: Computer ScienceComputer Science (R0)