Hindawi Publishing Corporation Journal of Discrete Mathematics Volume 2013, Article ID 692645, 4 pages http://dx.doi.org/10.1155/2013/692645
Research Article Another Note on Dilworth’s Decomposition Theorem Wim Pijls and Rob Potharst Erasmus University Rotterdam, P.O. Box 1738, 3000 DR Rotterdam, e Netherlands Correspondence should be addressed to Wim Pijls;
[email protected] Received 8 June 2012; Accepted 9 November 2012 Academic Editor: Stefan Richter Copyright © 2013 W. Pijls and R. Potharst. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. is paper proposes a new proof of Dilworth’s theorem. e proof is based upon the min�ow/maxcut property in �ow networks. In relation to this proof, a new method to �nd both a Dilworth decomposition and a maximal antichain is presented.
1. Introduction Several proofs are known for Dilworth’s theorem. is theorem says that, in a poset 𝑃𝑃, a maximal antichain and a minimal path cover have equal size. Shortly aer Dilworth’s seminal paper [1] a “Note” [2] was published containing an algorithmic proof, that is, a proof which also gives a method to �nd a combination of a maximal antichain and a minimal path cover. e other proofs [1, 3–5] are nonalgorithmic. e key issue in [2] is the relation between a minimal path cover and a maximal antichain in 𝑃𝑃 on the one hand and a maximal matching and a minimal vertex cover (in this order) in an associated bipartite graph 𝐵𝐵 on the other hand. Dilworth’s theorem is proved in [2] using König’s theorem stating that, in a bipartite graph, a maximal matching and a minimal vertex cover have equal size. e combination of a maximal matching and a minimal vertex cover in 𝐵𝐵 corresponds to a max�ow/mincut combination in a �ow network 𝐵𝐵′ akin to 𝐵𝐵. So the obvious algorithm for a minimal path cover along with a maximal antichain is executing a max�ow/mincut algorithm in 𝐵𝐵′ associated indirectly to 𝑃𝑃. In the current paper a shortcut is proposed between max�ow/mincut and an optimal path cover jointly with an antichain. To a given poset 𝑃𝑃 we associate a �ow network 𝑁𝑁 which is much simpler than graph 𝐵𝐵′ constructed via a matching/vertex cover instance. A similar idea for �nding a maximal antichain is found in [6]. However, the discussion in that paper was not connected with Dilworth’s theorem.
Other more complex algorithms in this domain can be found in [7–10]. For an application of the maximal antichain we refer to [11, 12].
2. Some Preliminaries A poset 𝑃𝑃(𝑉𝑉𝑉