Abstract
Because of the central role Dilworth’s decomposition theorem plays in the structure theory of (partially) ordered sets, it is important to consider it from a computational point of view. The theorem is proved in two steps. The first step is to prove the result for finite ordered sets. Dilworth’s original proof in [1] of the finite result is constructive, but the notion of computational complexity is not addressed. In 1956, Fulkerson [2] gave a network flow proof of Dilworth’s theorem. This proof allows the calculation of a Dilworth decomposition using any of the fast network flow algorithms. In particular, a decomposition can be calculated in O(n 3) time using the network flow algorithms in [3] or [8].
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
References
R.P. Dilworth, A decomposition theorem for partially ordered sets, Ann. of Math. (2) 51 (1950), 161–166. Reprinted in Chapter 1 of this volume.
D.R. Fulkerson, Note on Dilworth’s decomposition theorem for partially ordered sets, Proc. Amer. Math. Soc. 7 (1956), 701–702.
A.V. Karazanov, Determining the maximal flow in a network with the method of Preflows, Soviet Math. Dokl. 15 (1974), 434–437.
H.A. Kierstead, An effective version of Dilworth’s Theorem, Trans. Amer. Math. Soc. 268 (1981), 63–77.
H.A. Kierstead, A polynomial time approximation algorithm for Dynamic Storage Allocation.
H.A. Kierstead, G.F. McNulty, and W.T. Trotter, A theory of recursive dimension for ordered sets, Order 1 (1984), 67–82.
H.A. Kierstead and W.T. Trotter, An extremal problem in recursive combinatorics, Congressus Numerantium 33 (1981), 143–153.
V.M. Malhotra, M.P. Kumar, and S.N. Maheshwari, An O(v 3) algorithm for finding maximum flows in networks, Information Proc. Letters 7 (1978), 277–278.
E. Szemerédi and W.T. Trotter, private communication.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1990 Springer Science+Business Media New York
About this chapter
Cite this chapter
Kierstead, H. (1990). Effective Versions of the Chain Decomposition Theorem. In: Bogart, K.P., Freese, R., Kung, J.P.S. (eds) The Dilworth Theorems. Contemporary Mathematicians. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4899-3558-8_5
Download citation
DOI: https://doi.org/10.1007/978-1-4899-3558-8_5
Publisher Name: Birkhäuser, Boston, MA
Print ISBN: 978-1-4899-3560-1
Online ISBN: 978-1-4899-3558-8
eBook Packages: Springer Book Archive