Another Note on Dilworth's Decomposition Theorem

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Nov 9, 2012 - So the obvious algorithm for a minimal path cover along with a maximal antichain is executing a maxflow/mincut algorithm in. ′ associated.
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 aer 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