Abstract
On-line chain partition is a two-player game between Spoiler and Algorithm. Spoiler presents a partially ordered set, point by point. Algorithm assigns incoming points (immediately and irrevocably) to the chains which constitute a chain partition of the order. The value of the game for orders of width w is a minimum number val(w) such that Algorithm has a strategy using at most val(w) chains on orders of width at most w. We analyze the chain partition game for up-growing semi-orders. Surprisingly, the golden ratio comes into play and the value of the game is \(\big\lfloor\frac{1+\sqrt{5}}{2}\; w \big\rfloor\).
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Baier, P., Bosek, B., Micek, P.: On-line chain partitioning of up-growing interval orders. Order 24(1), 1–13 (2007)
Bosek, B., Felsner, S., Kloch, K., Krawczyk, T., Matecki, G., Micek, P.: On-line chain partitions of orders: a survey. Order (2011). doi:10.1007/ s11083-011-9197-1
Felsner, S.: On-line chain partitions of orders. Theor. Comput. Sci. 175(2), 283–292 (1997)
Kierstead, H.A.: An effective version of Dilworth’s theorem. Trans. Am. Math. Soc. 268(1), 63–77 (1981)
Kierstead, H.A.: Recursive ordered sets. In: Combinatorics and Ordered Sets, Arcata, Calif., 1985. Contemp. Math., vol. 57, pp. 75–102. Amer. Math. Soc., Providence, RI (1986)
Kierstead, H.A., Trotter, Jr., W.T.: An extremal problem in recursive combinatorics. In: Proceedings of the Twelfth Southeastern Conference on Combinatorics, Graph Theory and Computing, Vol. II, (Baton Rouge, La., 1981), vol. 33, pp. 143–153 (1981)
Möhring, R.H.: Computationally tractable classes of ordered sets. In: Algorithms and Order, pp. 105–193. Kluwer Acad. Publ., Dordrecht (1989)
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Felsner, S., Kloch, K., Matecki, G. et al. On-line Chain Partitions of Up-growing Semi-orders. Order 30, 85–101 (2013). https://doi.org/10.1007/s11083-011-9228-y
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DOI: https://doi.org/10.1007/s11083-011-9228-y