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First-Fit is Linear on Posets Excluding Two Long Incomparable Chains

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Abstract

A poset is \((\underline{r}+\underline{s})\)-free if it does not contain two incomparable chains of size r and s, respectively. We prove that when r and s are at least 2, the First-Fit algorithm partitions every \((\underline{r}+\underline{s})\)-free poset P into at most 8(r − 1)(s − 1)w chains, where w is the width of P. This solves an open problem of Bosek et al. (SIAM J Discrete Math 23(4):1992–1999, 2010).

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Authors and Affiliations

  1. Département d’Informatique, Université Libre de Bruxelles, Brussels, Belgium

    Gwenaël Joret

  2. Department of Mathematics, University of South Carolina, Columbia, SC, USA

    Kevin G. Milans

Authors
  1. Gwenaël Joret
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  2. Kevin G. Milans
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Correspondence to Kevin G. Milans.

Additional information

This work was supported in part by the Actions de Recherche Concertées (ARC) fund of the Communauté française de Belgique. Gwenaël Joret is a Postdoctoral Researcher of the Fonds National de la Recherche Scientifique (F.R.S.–FNRS).

Kevin G. Milans acknowledges support of the National Science Foundation through a fellowship funded by the grant “EMSW21-MCTP: Research Experience for Graduate Students” (NSF DMS 08-38434).

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Joret, G., Milans, K.G. First-Fit is Linear on Posets Excluding Two Long Incomparable Chains. Order 28, 455–464 (2011). https://doi.org/10.1007/s11083-010-9184-y

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